#72268147

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂⁹×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

10905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=h,c^2=j,d^2=j,e^2=i,f^2=i,b^a=bh,c^a=cg,c^b=cgj,d^a=dgj,d^b=dj,d^c=dj,e^a=eghj,e^b=egh,e^d=ej,f^a=fj,f^b=fghj,f^c=fj,f^d=fj,f^e=fi

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72268149, #72268175, #72268182, …
Fitting subgroup of	#72268149
maximal subgroup of	#72268149, #72268170, #72268180
maximal normal subgroup of	#72268149, #72268170, #72268180

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