#72261997

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 39344 conjugacy classes of subgroups; only the 29387 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)(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,31)(29,32);(1,3)(2,4)(5,8)(6,7)(17,19,22,27)(18,23,24,30)(20,28,29,26)(21,25,31,32);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,14)(15,16)(18,24)(20,29)(23,30)(26,28);(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,14)(13,15)(19,27)(20,29)(23,30)(25,32);(1,2,5,6)(3,7,8,4)(9,12,13,16)(10,14,15,11)(17,20,22,29)(18,25,24,32)(19,26,27,28)(21,30,31,23);(1,2)(3,4)(5,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,21)(18,26)(19,25)(20,23)(22,31)(24,28)(27,32)(29,30);(1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,14)(15,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,29)(31,32)

Polycyclic presentation:

2^10|b^2=h,e^2=j,c^a=ch,d^b=dhi,d^c=dhj,e^a=ehj,e^b=ehi,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, 2, F.8);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, 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*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
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);