#72262811

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 12556 conjugacy classes of subgroups; only the 29341 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=h,f^2=h,b^a=bh,c^b=chi,d^a=dh,d^c=dj,e^a=ej,f^a=fh,f^c=fi,f^e=fhj,g^a=gij,g^b=gh,g^f=gj

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318222
maximal normal subgroup of	#72318222
maximal quotient of	#72262832

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