#72227117

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#1368
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	C₂×#346	C₂⁴
prime radical	C₂×#135	#106
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#443 ◁ #493217 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1185 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=eij,c^2=ej,d^2=g,e^2=h,f^2=ik,g^2=k,i^2=j,j^2=k,b^a=bf,c^a=ci,c^b=cfk,d^b=dk,d^c=dg,e^b=ej,e^c=ek,f^a=fi,f^b=fi,f^c=fi,f^e=fk,g^c=gk,h^b=hk,i^a=ij,i^b=ij,i^c=ij,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227121, #72227158, #72227175
maximal normal subgroup of	#72227121, #72227158, #72227175

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