#72261985

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 38779 conjugacy classes of subgroups; only the 29385 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=h,e^2=j,b^a=bh,c^a=ch,d^a=dhi,d^c=dhj,e^a=eh,e^b=ehij,e^c=ehi,e^d=ehi,f^d=fhij,f^e=fj,g^b=gh,g^d=gj

Conjugacy classes & character table

160 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.8);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);