#72304138

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

16876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=g,c^2=hj,d^2=ij,c^a=cgh,c^b=ch,d^a=dgi,d^b=dghi,e^c=egh,e^d=ej,f^a=fi,f^c=fgj,f^d=fi

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304145
maximal normal subgroup of	#72304145

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