#503525

order	512
nilpotency class	4
isoclinism	isoclinic class of #501058
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

813 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 813 conjugacy classes of subgroups — too many to draw. Only its 178 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)(11,21,14,31)(12,25,29,32)(13,30,23,18)(16,27,19,24)(17,22,34,26)(28,35,36,39)(33,40,38,37);(1,3,5,4)(2,6,8,7)(9,11,13,24,15,14,23,27)(10,16,18,21,20,19,30,31)(12,26,28,40,29,22,36,37)(17,32,33,35,34,25,38,39);(1,3,5,4)(2,7,8,6)(9,12,27,40,13,28,11,22,15,29,24,37,23,36,14,26)(10,17,31,39,18,33,16,32,20,34,21,35,30,38,19,25);(1,4,5,3)(2,7,8,6)(9,11,23,27,15,14,13,24)(10,16,30,31,20,19,18,21)(12,22,36,40,29,26,28,37)(17,32,38,39,34,25,33,35)

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=ef,d^2=ei,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,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, 410721);