#72263221

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 8210 conjugacy classes of subgroups; only the 3065 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=h,d^2=h,f^2=ij,c^b=ch,d^a=dj,d^c=dg,e^a=ej,e^b=eghj,e^c=ej,e^d=ei,f^a=fhij,f^c=fij,f^d=fg,f^e=fi

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263231
maximal normal subgroup of	#72263231

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