#72280205

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	#11022
commutator	C₂³×C₈	C₂²×C₄
Frattini, cosocle	C₂⁴×C₈	C₂³
prime radical	C₂×#33517	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ C₂×#1752 ◁ G
LECS	G ▷ C₂⁴×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|b^2=j,c^2=d,d^2=ij,e^2=ij,f^2=j,i^2=j,b^a=be,c^a=ceg,c^b=cefj,d^a=di,d^b=dij,e^a=ei,e^b=ei,e^c=eh,f^a=fj,f^b=fj,f^c=fhj,g^b=ghj,g^c=ghj,i^a=ij,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280299
maximal normal subgroup of	#72280299

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