#72234041

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 9759 conjugacy classes of subgroups; only the 3029 normal subgroups are shown.

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:

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

Polycyclic presentation:

2^10|a^2=g,f^2=i,b^a=bh,c^b=cg,d^a=dghi,d^c=dj,e^a=ej,e^c=ehi,f^a=fj,f^c=fhij,f^d=fi

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234056
maximal normal subgroup of	#72234056

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