#501561

order	512
nilpotency class	4
isoclinism	isoclinic class of #501014
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×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

531 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=ei,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=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, 408757);