#512313

order	512
nilpotency class	4
isoclinism	isoclinic class of #511960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
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

564 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)(12,24)(13,25)(16,26)(17,33)(18,31)(19,30)(21,23)(22,37)(27,34)(28,29)(32,36)(35,38);(1,3,4,7)(2,5,6,8)(9,10,16,31,13,18,12,15,14,19,24,29,25,28,26,30)(11,17,32,40,22,33,21,27,23,35,37,39,36,38,20,34);(2,6)(5,8);(9,11,12,21,25,36,16,32,14,23,26,20,13,22,24,37)(10,17,15,27,28,38,31,40,19,35,30,34,18,33,29,39)

Polycyclic presentation:

2^9|b^2=efhi,d^2=e,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,h^a=hi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419509);