#503029

order	512
nilpotency class	4
isoclinism	isoclinic class of #501132
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂³	C₂×#120
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

728 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)(11,21)(13,26)(16,30)(18,27)(22,35)(23,36)(24,38)(25,29)(28,37)(31,39)(32,34)(33,40);(2,5)(6,7)(9,10,15,20)(11,22,24,33)(12,19,14,17)(13,27,26,18)(16,28,31,23)(21,35,38,40)(25,32,29,34)(30,37,39,36);(9,11,13,21,15,24,26,38)(10,16,18,30,20,31,27,39)(12,23,25,36,14,28,29,37)(17,22,32,35,19,33,34,40);(1,3,4,8)(2,6,5,7)(9,12,15,14)(10,17,20,19)(11,23,24,28)(13,25,26,29)(16,22,31,33)(18,32,27,34)(21,36,38,37)(30,35,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410225);