#506225

order	512
nilpotency class	4
isoclinism	isoclinic class of #500676
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33519	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

725 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=eg,d^2=i,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,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, 413421);