#492516

order	512
nilpotency class	4
isoclinism	isoclinic class of #492512
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19971	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

494 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 256

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

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

Polycyclic presentation:

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

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, 399712);