#250096

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
prime radical	#24230	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1145 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:

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

Polycyclic presentation:

2^9|b^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, 157292);