#503051

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

731 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:

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

Polycyclic presentation:

2^9|a^2=ei,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, 410247);