#72249479

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 6899 conjugacy classes of subgroups; only the 29669 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|b^2=i,e^2=h,b^a=bi,c^a=ci,e^a=ei,e^c=ej,e^d=ei,f^a=fj,f^b=fhj,f^e=fj,g^b=gi,g^e=gj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249567, #72255883
maximal normal subgroup of	#72249567, #72255883

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, 2, F.9);;
gap> SetPower(coll, 5, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*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);