#72232662

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

20122 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:

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

Polycyclic presentation:

2^10|a^2=h,c^2=g,f^2=hi,c^a=cg,c^b=ci,d^a=dj,e^a=eh,f^a=fgi,f^b=fgh,f^c=fij,f^d=fgi,f^e=fhij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232710, #72233469, #72327788
maximal normal subgroup of	#72232710, #72233469, #72327788

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