#72287678

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72285235
commutator	C₂×M₃₂	C₂³×C₄
Frattini, cosocle	C₄×M₃₂	C₂⁴
prime radical	#3444	SD₁₆
LCS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ #7622661 ◁ G
LECS	G ▷ C₄×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1559 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,14)(4,10)(5,24)(6,27)(7,13)(8,11)(9,18)(12,22)(15,44)(16,17)(19,46)(20,38)(21,43)(23,37)(25,39)(26,45)(28,40)(29,50)(30,41)(31,33)(32,47)(34,53)(35,55)(36,56)(42,61)(48,51)(49,58)(52,54)(57,64)(59,63)(60,62);(1,2,4,10)(3,15,17,48,20,49,18,50)(5,11,8,24)(6,28)(7,33,23,45)(9,29,14,44,16,51,38,58)(12,40)(13,26,37,31)(19,53,55,63,57,36,54,43)(21,46,34,35,59,64,56,52)(22,60)(25,42)(27,62)(30,47)(32,39)(41,61);(1,3,7,34,8,18,31,21,4,20,23,56,5,17,26,59)(2,9,13,43,11,38,33,36,10,16,37,63,24,14,45,53)(6,29,61,55,30,44,62,57,22,51,32,54,25,58,40,19)(12,15,47,64,39,48,28,52,27,49,42,46,41,50,60,35);(1,2)(3,16)(4,10)(5,24)(6,27)(7,13)(8,11)(9,20)(12,22)(14,18)(15,29)(17,38)(19,35)(21,53)(23,37)(25,39)(26,45)(28,40)(30,41)(31,33)(32,47)(34,63)(36,59)(42,61)(43,56)(44,48)(46,54)(49,51)(50,58)(52,57)(55,64)(60,62)

Polycyclic presentation:

2^11|b^2=e,c^2=ehik,e^2=h,f^2=k,g^2=ij,i^2=j,j^2=k,b^a=bhk,c^a=cf,c^b=cfgik,d^a=dh,d^b=dk,d^c=dk,e^c=ej,f^a=fk,f^b=fh,f^c=fk,f^d=fk,g^a=ghk,g^b=ghijk,g^c=gijk,g^d=gk,g^e=gk,g^f=gk,h^c=hk,i^b=ij,i^c=ij,j^b=jk,j^c=jk

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287845, #72343265
maximal normal subgroup of	#72287845, #72343265

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.5*F.8*F.9*F.11);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.11);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.7*F.9*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);