#506321

order	512
nilpotency class	4
isoclinism	isoclinic class of #500843
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

494 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:

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

Polycyclic presentation:

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

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, 413517);