#500241

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	#45441	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1105 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)(9,10,15,19)(11,16,13,26)(12,22,25,37)(14,18,28,34)(17,31,32,24)(20,35,23,38)(21,30,27,33)(29,39,40,36);(2,5)(6,8)(9,11,15,13)(10,16,19,26)(12,20,25,23)(14,27,28,21)(17,29,32,40)(18,33,34,30)(22,35,37,38)(24,36,31,39);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,20)(13,23)(14,24)(15,25)(16,29)(18,22)(19,32)(21,36)(26,40)(27,39)(28,31)(30,35)(33,38)(34,37);(2,5)(6,8)(9,13,15,11)(10,16,19,26)(12,23,25,20)(14,27,28,21)(17,29,32,40)(18,30,34,33)(22,35,37,38)(24,39,31,36)

Polycyclic presentation:

2^9|a^2=i,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, 407437);