#242690

order	512
nilpotency class	3
isoclinism	isoclinic class of #242681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

687 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,g^b=gh,d^c=dh,e^c=eh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149886);