#72332476

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

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

63985 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(20,24)(22,23);(9,10)(11,14)(12,13)(15,16);(5,6)(7,8)(11,15)(14,16)(17,18,21,19)(20,23,24,22);(2,4)(5,6,7,8)(10,13)(11,15)(17,19,21,18)(20,23,24,22);(9,11)(10,14)(12,15)(13,16);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,22)(19,23)(21,24)

Polycyclic presentation:

2^10|c^2=j,d^2=ij,d^b=dg,d^c=di,e^c=eg,e^d=eg,f^a=fj,f^c=fj,f^d=fh

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72332480
maximal normal subgroup of	#72332480
maximal quotient of	#72333331, #72336469

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