#72342342

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 14255 conjugacy classes of subgroups; only the 29393 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:

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

Polycyclic presentation:

2^10|a^2=i,d^2=j,f^2=hj,b^a=bi,d^a=dh,d^c=dj,e^a=ei,e^b=ehj,e^d=ej,f^b=fhj,f^c=fhj,f^d=fj,g^a=gj,g^b=gj,g^e=gi,g^f=gij

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342349
maximal normal subgroup of	#72342349

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