#72343025

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72320558
commutator	C₂²×C₁₆	C₂³×C₄
Frattini, cosocle	C₂×#504	C₂⁴
prime radical	#72216688	C₂
LCS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ #7622661 ◁ G
LECS	G ▷ C₂×#504 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5384 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:

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

Polycyclic presentation:

2^11|c^2=fgi,d^2=hk,f^2=ij,g^2=ij,i^2=jk,j^2=k,b^a=bh,c^a=cf,c^b=ce,d^a=dk,d^c=djk,e^c=eh,f^a=fi,f^b=fk,f^c=fi,f^d=fk,g^a=gi,g^b=ghk,g^c=gi,g^d=gk,g^e=gk,h^c=hk,i^a=ij,i^c=ij,j^a=jk,j^c=jk

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343072, #72343265, #72343292
maximal normal subgroup of	#72343072, #72343265, #72343292

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, 3, F.6*F.7*F.9);;
gap> SetPower(coll, 4, F.8*F.11);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.10*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);