#72249493

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 39882 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:

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

Polycyclic presentation:

2^10|b^2=i,f^2=j,b^a=bi,c^a=ci,d^a=di,e^c=ej,e^d=eh,f^a=fij,f^b=fhij,f^d=fj,f^e=fhi,g^b=gi,g^e=gj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249567
maximal normal subgroup of	#72249567

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