#72336767

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 9615 conjugacy classes of subgroups; only the 3049 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gj,d^2=h,f^2=hi,b^a=bgh,c^a=cg,c^b=cgi,d^a=dgh,d^b=dghj,d^c=dh,e^a=eh,e^b=eh,e^c=eh,e^d=eh,f^a=fj,f^c=fgj,f^d=fhi

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336887
maximal normal subgroup of	#72336887

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