#72233145

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#10980
commutator	C₂⁶	C₂²×C₄
Frattini, cosocle	C₂⁴×D₄	C₂³
prime radical	#688032	C₂
LCS	G ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂×D₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

4668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^10|b^2=fi,c^2=e,b^a=bf,c^a=cd,c^b=cg,d^b=dh,d^c=di,e^a=ei,e^d=ej,f^c=fhj,g^a=gj,g^b=gh,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233148, #72233155, #72233185
maximal normal subgroup of	#72233148, #72233155, #72233185

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