#250086

order	512
nilpotency class	3
isoclinism	isoclinic class of #250071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,e^b=eh,g^b=gi,e^c=ei,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157282);