#72333463

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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 9735 conjugacy classes of subgroups; only the 3589 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)(3,4)(6,8);(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(18,22)(19,25)(20,28)(31,32);(1,2,3,4)(10,11)(14,15)(18,22)(23,26)(24,30)(31,32);(2,4)(10,11)(12,16)(17,18,21,22)(19,26,25,23)(20,24,28,30)(27,32,29,31);(1,2,3,4)(9,11,13,10)(12,14,16,15)(17,19)(18,23)(20,29)(21,25)(22,26)(24,32)(27,28)(30,31);(1,2)(3,4)(9,12)(10,14)(11,15)(13,16)(17,20)(18,24)(19,27)(21,28)(22,30)(23,31)(25,29)(26,32)

Polycyclic presentation:

2^10|c^2=i,d^2=gh,e^2=h,b^a=bj,c^a=ci,c^b=ch,d^a=di,d^b=dg,d^c=dghi,e^a=ei,e^b=eg,e^c=ehi,e^d=eg,f^b=fgh,f^c=fi,f^d=fh,f^e=fg

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333496
maximal normal subgroup of	#72333496

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