#503109

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3147	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1306 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1306 conjugacy classes of subgroups — too many to draw. Only its 202 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410305);