#72280830

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

21891 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=i,c^2=hi,d^2=i,f^2=gi,b^a=bi,c^a=ch,c^b=ch,d^a=di,d^c=dghi,e^a=eh,e^b=eij,e^c=eghi,e^d=eh,f^b=fi,f^e=fgi

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

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