#72265967

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 19856 conjugacy classes of subgroups; only the 29349 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 32 points:

(1,2,4,6)(3,7,8,5)(18,22)(20,28)(23,26)(27,31);(3,8)(5,7)(11,15)(14,16);(10,13)(14,16)(20,28)(24,29)(27,31)(30,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(18,22)(19,25)(24,29)(27,31);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30);(1,3)(2,5)(4,8)(6,7)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32);(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)

Polycyclic presentation:

2^10|a^2=i,e^2=j,d^a=di,d^c=dh,e^a=eij,e^b=eh,e^d=ej,f^a=fi,f^b=fi,f^d=fj,f^e=fj,g^a=gj,g^c=gj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265968
maximal normal subgroup of	#72265968
maximal quotient of	#72266583, #72266584

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