#509201

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

782 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 148 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=i,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=dei,f^c=fh,h^c=hi,f^d=fh,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, 416397);