#72264397

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#392
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	C₂⁴×D₄	D₄
LCS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

5861 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=fj,c^2=f,g^2=j,b^a=bg,c^a=ch,c^b=ce,d^c=df,e^a=ei,g^a=gj,g^b=gj,h^b=hi

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264401, #72264427, #72264634
maximal normal subgroup of	#72264401, #72264427, #72264634

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