#500233

order	512
nilpotency class	4
isoclinism	isoclinic class of #499803
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	#2431	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1052 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)(9,10,15,13)(11,16,21,26)(12,22,25,37)(14,18,29,27)(17,32,23,24)(19,34,36,40)(20,31,28,33)(30,39,38,35);(2,5)(6,8)(9,11,15,21)(10,16,13,26)(12,19,25,36)(14,28,29,20)(17,30,23,38)(18,33,27,31)(22,34,37,40)(24,35,32,39);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,19)(13,23)(14,24)(15,25)(16,30)(18,22)(20,35)(21,36)(26,38)(27,37)(28,39)(29,32)(31,34)(33,40);(2,5)(6,8)(9,13,15,10)(11,16,21,26)(12,23,25,17)(14,27,29,18)(19,30,36,38)(20,31,28,33)(22,24,37,32)(34,39,40,35)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=e,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,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, 407429);