#72318322

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷×C₄

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

45517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|d^2=hij,e^2=hi,c^a=ch,c^b=ci,d^b=dhij,e^a=ehi,e^b=ehi,f^a=fij,f^b=fij,g^b=gi,g^c=gj,g^f=gj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318445, #72319411
maximal normal subgroup of	#72318445, #72319411

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.8*F.9*F.10);;
gap> SetPower(coll, 5, F.8*F.9);;
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.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, 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);