#72306597

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂⁷×C₄⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

22619 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^10|e^2=j,f^2=j,b^a=bi,c^b=cg,d^a=dgi,d^c=di,e^a=ej,e^b=ei,e^c=eh,e^d=eij,f^a=fhj,f^b=fj,f^c=fj,f^d=fi,f^e=fj

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306602, #72324494
maximal normal subgroup of	#72306602, #72324494

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