#508267

order	512
nilpotency class	4
isoclinism	isoclinic class of #508066
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	C₂×#127	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

801 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415463);