#492518

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	#19975	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 507 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)(9,10,15,19)(11,20,21,39)(12,25,17,37)(13,28,31,27)(14,32,30,18)(16,34,35,23)(22,40,36,29)(24,26,33,38);(9,11,15,21)(10,16,19,35)(12,26,17,38)(13,29,31,40)(14,33,30,24)(18,22,32,36)(20,27,39,28)(23,37,34,25);(2,5)(3,8)(6,7)(11,21)(16,35)(20,39)(22,36)(23,34)(24,33)(26,38)(29,40);(9,12,15,17)(10,13,19,31)(11,22,21,36)(14,28,30,27)(16,24,35,33)(18,37,32,25)(20,34,39,23)(26,40,38,29)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^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, 399714);