#244099

order	512
nilpotency class	3
isoclinism	isoclinic class of #238010
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#400
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

555 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=ef,c^2=e,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gi,d^c=di,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151295);