#72309489

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄²×D₄
commutator	C₄²	C₂²×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
socle	C₂²	D₄×#97
prime radical	C₂²×D₄	C₂×M₁₆
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#347 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

991 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=h,c^2=e,d^2=g,f^2=h,g^2=i,i^2=j,b^a=bf,d^b=dh,d^c=di,e^d=ej,f^a=fh,f^b=fh,g^c=gj

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309511, #72309909, #72309999, …
maximal normal subgroup of	#72309511, #72309909, #72309999, …

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