#72342496

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×#106
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂²×D₄	C₂×SD₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×SD₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

891 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)(3,8)(6,7)(11,17)(15,20)(18,21)(19,24);(2,6)(3,8)(5,7)(10,14)(11,18,17,21)(12,16)(15,19,20,24)(22,23);(1,2,4,7)(3,6,8,5)(9,10,13,14)(11,15,17,20)(12,16,22,23)(18,24,21,19);(1,2,3,5,4,7,8,6)(9,11,10,15,13,17,14,20)(12,21,16,19,22,18,23,24)

Polycyclic presentation:

2^10|b^2=e,c^2=hj,d^2=efi,f^2=h,g^2=i,i^2=j,c^a=cf,c^b=cfhj,d^a=dfj,d^b=dfgij,d^c=df,e^d=ej,f^a=fh,f^b=fh,f^c=fh,g^b=gi,g^d=gi,i^b=ij,i^d=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342502, #72342508, AGL(1,ℤ/8)×#365
maximal normal subgroup of	#72342502, #72342508, AGL(1,ℤ/8)×#365

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