#503531

order	512
nilpotency class	4
isoclinism	isoclinic class of #501076
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33190	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

790 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410727);