#503297

order	512
nilpotency class	4
isoclinism	isoclinic class of #501072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
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

724 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)(11,17)(12,24)(13,27)(14,18)(15,26)(16,19)(20,21)(22,33)(23,35)(25,36)(28,29)(30,32)(31,34)(37,40)(38,39);(1,3,5,4)(2,6,8,7)(9,11)(10,17)(12,25)(13,21)(14,31)(15,22)(16,23)(18,34)(19,35)(20,27)(24,36)(26,33)(28,37)(29,40)(30,39)(32,38);(9,12,24,10,15,27,30,18,16,28,29,19,14,32,13,26)(11,20,36,34,22,37,39,35,23,38,40,33,31,25,21,17);(1,4,5,3)(2,7,8,6)

Polycyclic presentation:

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

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, 410493);