#501752

order	512
nilpotency class	4
isoclinism	isoclinic class of #501052
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 408948);