#72237301

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

445307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 8 generators on 32 points:

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

Polycyclic presentation:

2^10|d^2=i,e^2=ij,c^b=ci,d^a=di,d^c=dij,e^a=eij,e^b=ej,f^b=fi,f^e=fij,g^b=gj,g^c=gj,h^b=hi,h^c=hi

Conjugacy classes & character table

268 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262013
maximal normal subgroup of	#72262013

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