#511994

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

974 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)(9,10,15,21)(11,22,25,38)(12,26,30,37)(13,20,14,19)(16,32,35,23)(17,31,36,24)(18,28,33,29)(27,39,40,34);(1,3,4,8)(2,5,6,7);(9,11)(10,16)(12,27)(13,24)(14,31)(15,25)(17,33)(18,36)(19,34)(20,39)(21,35)(22,26)(23,29)(28,32)(30,40)(37,38);(9,11)(10,17)(12,23)(13,31)(14,24)(15,25)(16,33)(18,35)(19,38)(20,22)(21,36)(26,39)(27,29)(28,40)(30,32)(34,37)

Polycyclic presentation:

2^9|a^2=ei,b^2=e,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, 419190);