#500237

order	512
nilpotency class	4
isoclinism	isoclinic class of #499795
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	#45442	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1121 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=e,d^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,f^b=fh,h^c=hi,g^f=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, 407433);