#72343170

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#393031
commutator	C₂×#131	C₂³×C₄
Frattini, cosocle	C₄×#131	C₂⁴
socle	C₂	#72343138
prime radical	#72217287	C₂
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

2765 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:

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

Polycyclic presentation:

2^11|a^2=e,c^2=fgk,d^2=hi,e^2=i,h^2=j,i^2=k,j^2=k,b^a=bk,c^a=cf,c^b=cfg,d^a=di,d^b=dijk,d^c=dh,e^d=ek,f^b=fijk,f^c=fij,f^d=fk,g^a=gij,g^b=gij,g^c=gij,g^f=gk,h^a=hk,h^c=hj,j^c=jk

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343240, #72343364, #72343423
maximal normal subgroup of	#72343240, #72343364, #72343423

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