#72336772

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 9939 conjugacy classes of subgroups; only the 3033 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)(4,6)(5,8)(9,10,13,15)(11,14,16,12)(18,22)(20,23)(25,26)(27,31)(28,30)(29,32);(1,2,4,6)(3,5,8,7)(9,11,13,16)(10,14,15,12)(17,18,21,22)(19,23,24,20)(25,27,30,32)(26,31,28,29);(3,8)(5,7)(10,15)(12,14)(18,22)(20,23)(25,26)(27,31)(28,30)(29,32);(3,8)(5,7)(26,28)(27,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,24)(22,23)(25,28,30,26)(27,29,32,31);(1,3,4,8)(2,5,6,7)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,23)(22,24)(25,29)(26,27)(28,32)(30,31)

Polycyclic presentation:

2^10|a^2=g,b^2=gj,e^2=ghj,f^2=hi,b^a=bgh,c^a=cg,c^b=cgi,d^a=dghj,d^b=dghj,d^c=dghj,e^a=ehj,e^b=eh,e^c=eghj,e^d=eghj,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, 5, F.7*F.8*F.10);;
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*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*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.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8*F.10);;
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);