#500249

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1024 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)(3,6)(4,5)(7,8)(9,10,14,13)(11,19,21,16)(12,23,25,39)(15,18,28,27)(17,32,24,26)(20,35,36,38)(22,33,29,31)(30,37,34,40);(2,5)(6,8)(9,11,14,21)(10,16,13,19)(12,20,25,36)(15,29,28,22)(17,30,24,34)(18,33,27,31)(23,38,39,35)(26,37,32,40);(1,3,4,7)(2,6,5,8)(9,12,14,25)(10,17,13,24)(11,20,21,36)(15,26,28,32)(16,30,19,34)(18,23,27,39)(22,37,29,40)(31,38,33,35);(2,5)(6,8)(9,13,14,10)(11,16,21,19)(12,24,25,17)(15,27,28,18)(20,30,36,34)(22,31,29,33)(23,26,39,32)(35,37,38,40)

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,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, 407445);