#72333467

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

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

49457 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:

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

Polycyclic presentation:

2^10|a^2=i,b^a=bg,c^a=ci,c^b=cgh,d^a=dgh,e^a=ehj,f^a=fi,f^b=fhj,f^d=fgj,f^e=fgh

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333496
maximal normal subgroup of	#72333496

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