#72232698

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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

26574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|c^2=g,d^2=ij,f^2=hj,c^a=cg,c^b=ci,d^b=dgh,d^c=dij,e^d=eij,f^a=fj,f^d=fgi,f^e=fh

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232710, #72233474, #72327940, …
maximal normal subgroup of	#72232710, #72233474, #72327940, …
maximal quotient of	#72331539

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