#500038

order	512
nilpotency class	4
isoclinism	isoclinic class of #499831
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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×#585	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1558 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)(3,7)(4,6)(5,8)(9,10,14,18)(11,15,21,29)(12,16,22,30)(13,24,25,17)(19,27,26,32)(20,28,35,40)(23,37,36,31)(33,38,34,39);(1,3,4,8)(2,5,6,7)(11,19)(12,22)(13,25)(15,27)(16,30)(17,24)(20,34)(21,26)(28,39)(29,32)(33,35)(38,40);(9,11)(10,15)(12,20)(13,26)(14,21)(16,28)(17,32)(18,29)(19,25)(22,35)(23,34)(24,27)(30,40)(31,39)(33,36)(37,38);(2,6)(5,7)(9,12,14,22)(10,16,18,30)(11,20,21,35)(13,23,25,36)(15,28,29,40)(17,31,24,37)(19,33,26,34)(27,38,32,39)

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=gi,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, 407234);