#72319378

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 20830 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:

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

Polycyclic presentation:

2^10|d^2=j,c^a=ch,c^b=ci,d^b=dj,d^c=dj,e^d=ehi,f^a=fij,f^b=fij,g^b=gi,g^c=gj,g^f=gj

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319433
maximal normal subgroup of	#72319433

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