#486296

order	512
nilpotency class	4
isoclinism	isoclinic class of #486291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

903 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 903 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gi,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393492);