#489042

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
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	C₄×C₈	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 719 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:

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

Polycyclic presentation:

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

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, 396238);