#72232685

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 9517 conjugacy classes of subgroups; only the 3009 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=hj,c^2=gh,e^2=j,f^2=hi,b^a=bj,c^a=cg,c^b=ci,d^a=dj,d^b=dj,d^c=dj,e^a=ehj,e^b=eh,e^c=eh,e^d=ej,f^a=fgij,f^b=fhij,f^c=fgj,f^d=fgij,f^e=fghj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232710
maximal normal subgroup of	#72232710

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