#505413

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

453 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)(9,10,14,18)(11,20,23,16)(12,17,27,29)(13,30,19,15)(21,32,37,36)(22,31,24,33)(25,38,39,34)(26,28,40,35);(2,5)(6,7)(9,11,10,16,14,23,18,20)(12,25,29,38,27,39,17,34)(13,22,15,31,19,24,30,33)(21,26,36,28,37,40,32,35);(9,12,14,27)(10,17,18,29)(11,21,23,37)(13,28,19,35)(15,26,30,40)(16,32,20,36)(22,34,24,38)(25,31,39,33);(1,3,4,8)(2,6,5,7)(9,13,10,15,14,19,18,30)(11,22,16,31,23,24,20,33)(12,26,29,28,27,40,17,35)(21,25,36,38,37,39,32,34)

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=i,d^2=efhi,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, 412609);