#72319552

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#182043
commutator	C₂²×C₈	C₂²×C₄²
Frattini, cosocle	C₄×M₃₂	C₂⁴
socle	C₂	#72285281
prime radical	#3444	M₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #346 ◁ #479908 ◁ G
LECS	G ▷ C₄×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1767 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:

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

Polycyclic presentation:

2^11|a^2=h,b^2=e,c^2=fgijk,e^2=h,f^2=k,g^2=i,i^2=j,j^2=k,c^a=cf,c^b=cijk,d^a=dh,d^c=dhk,e^c=ejk,f^c=fh,f^d=fk,g^a=ghk,g^b=gj,g^c=gh,g^e=gk,g^f=gk,h^c=hk,i^b=ik

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319914, #72343465, #72343482
maximal normal subgroup of	#72319914, #72343465, #72343482

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