#505412

order	512
nilpotency class	4
isoclinism	isoclinic class of #501868
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1771 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

461 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412608);