#72267407

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

23369 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|c^2=hi,d^2=gi,e^2=j,f^2=j,c^a=ch,c^b=cj,d^a=dj,d^b=dg,d^c=di,e^d=ei,f^c=fi,f^e=fj

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267409
maximal normal subgroup of	#72267409
maximal quotient of	#72267464

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