#499502

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#466
prime radical	C₂×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

575 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,15,27,26,37,25,17,14,20,12,18,13,19,29,36)(11,16,28,30,23,32,38,35,24,34,22,31,21,33,39,40);(2,5)(6,8)(9,11)(10,16)(12,22)(13,21)(14,24)(15,28)(17,35)(18,31)(19,33)(20,34)(23,26)(25,38)(27,30)(29,39)(32,37)(36,40);(10,17)(11,21)(12,25)(13,26)(15,29)(16,30)(18,19)(20,36)(22,28)(23,24)(27,37)(31,34)(32,40)(33,35);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406698);