#503294

order	512
nilpotency class	4
isoclinism	isoclinic class of #501058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

747 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 747 conjugacy classes of subgroups — too many to draw. Only its 190 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)(9,10,13,18)(11,20,15,16)(12,23,26,38)(14,29,28,31)(17,32,33,27)(19,22,35,25)(21,36,24,39)(30,37,34,40);(1,3,5,4)(2,6,8,7)(9,11,13,15)(10,16,18,20)(12,24,26,21)(14,25,28,22)(17,30,33,34)(19,29,35,31)(23,36,38,39)(27,37,32,40);(9,12,25,40,15,24,28,32,13,26,22,37,11,21,14,27)(10,17,29,36,20,34,35,38,18,33,31,39,16,30,19,23);(1,4,5,3)(2,7,8,6)(10,18)(16,20)(17,33)(19,35)(23,38)(29,31)(30,34)(36,39)

Polycyclic presentation:

2^9|a^2=e,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,g^a=gi,h^a=hi,c^b=cg,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, 410490);