#507509

order	512
nilpotency class	4
isoclinism	isoclinic class of #501764
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

454 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eg,c^2=ef,d^2=ei,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,g^c=gi

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, 414705);