#502730

order	512
nilpotency class	4
isoclinism	isoclinic class of #501172
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×D₄	C₂⁴
prime radical	#3485	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

940 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)(9,10,14,18)(11,19,20,32)(12,23,24,16)(13,17,26,31)(15,27,29,22)(21,35,34,33)(25,38,39,30)(28,36,37,40);(1,3,4,8)(2,5,6,7)(11,20)(12,24)(16,23)(19,32)(25,39)(28,37)(30,38)(36,40);(9,11,13,22)(10,15,17,19)(12,21,25,36)(14,20,26,27)(16,28,30,33)(18,29,31,32)(23,37,38,35)(24,34,39,40);(2,6)(5,7)(9,12,14,24)(10,16,18,23)(11,21,20,34)(13,25,26,39)(15,28,29,37)(17,30,31,38)(19,33,32,35)(22,36,27,40)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^c=gh,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, 409926);