#503552

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 753 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=efh,d^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,g^c=gi

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, 410748);