#512298

order	512
nilpotency class	4
isoclinism	isoclinic class of #511890
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

609 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)(3,5)(4,6)(7,8)(10,15)(11,20)(12,16)(13,24)(17,27)(18,33)(19,32)(21,36)(22,25)(23,37)(26,30)(28,34)(31,40)(35,39);(1,3,4,7)(2,5,6,8)(9,10,16,27,14,17,12,15)(11,18,22,35,23,31,21,28)(13,19,30,38,26,32,24,29)(20,34,36,40,37,39,25,33);(2,6)(5,8)(10,17)(15,27)(18,31)(19,32)(28,35)(29,38)(33,40)(34,39);(9,11,12,21,14,23,16,22)(10,18,15,28,17,31,27,35)(13,25,24,37,26,36,30,20)(19,33,29,39,32,40,38,34)

Polycyclic presentation:

2^9|b^2=efi,d^2=e,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,g^d=ghi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419494);