#502698

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	#3487	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

903 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)(3,7)(4,6)(5,8)(9,10,15,19)(11,20,24,37)(12,17,14,28)(13,26,27,18)(16,30,32,23)(21,35,22,36)(25,34,29,33)(31,38,40,39);(1,3,4,8)(2,5,6,7)(11,21)(12,14)(16,31)(17,28)(20,35)(22,24)(23,39)(25,29)(30,38)(32,40)(33,34)(36,37);(9,11,13,23,15,24,27,30)(10,16,18,20,19,32,26,37)(12,22,25,39,14,21,29,38)(17,31,33,36,28,40,34,35);(2,6)(5,7)(9,12,15,14)(10,17,19,28)(11,22,24,21)(13,25,27,29)(16,31,32,40)(18,33,26,34)(20,36,37,35)(23,39,30,38)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=i,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, 409894);