#267485

order	512
nilpotency class	3
isoclinism	isoclinic class of #267467
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25739	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1009 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:

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

Polycyclic presentation:

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

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, 174681);