#72312133

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

46773 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^10|e^2=h,c^b=ch,d^a=di,d^c=dh,e^a=ehij,e^b=ej,e^c=ej,e^d=eh,f^a=fij,g^d=gij,g^e=gij,g^f=gj

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312142, #72341497, #72341503
maximal normal subgroup of	#72312142, #72341497, #72341503

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