#72334250

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹

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

18525 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=ij,b^a=bi,c^a=cij,c^b=ch,d^b=dghij,d^c=dghij,e^b=ehij,e^c=ehij,f^b=fghi,f^c=fij,f^d=fhi,f^e=fg

Conjugacy classes & character table

118 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, 2, F.7);;
gap> SetPower(coll, 3, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);