#72340331

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 12363 conjugacy classes of subgroups; only the 29389 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)(3,5)(4,7)(6,8)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32);(9,10)(11,13)(12,14)(15,16)(21,29)(24,26)(27,32)(30,31);(18,19)(23,25)(24,26)(30,31);(1,3,4,8)(2,5,7,6)(9,10,12,14)(11,13,16,15);(1,3,4,8)(2,6,7,5)(9,10,12,14)(11,15,16,13)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32);(1,2)(3,5)(4,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)

Polycyclic presentation:

2^10|d^2=j,e^2=ij,c^a=ci,d^b=dh,e^a=ehij,e^b=eh,e^c=ei,f^a=fi,f^d=fj,f^e=fj,g^b=gi,g^e=gj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340343, #72340369, #72340376
maximal normal subgroup of	#72340343, #72340369, #72340376

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