#511996

order	512
nilpotency class	4
isoclinism	isoclinic class of #511755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#576	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

966 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,h^a=hi,f^c=fh,h^c=hi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419192);