#512350

order	512
nilpotency class	4
isoclinism	isoclinic class of #512070
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

500 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efi,d^2=eh,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=gh,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, 419546);