#489058

order	512
nilpotency class	4
isoclinism	isoclinic class of #488912
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	#33580	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

744 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 154 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)(11,21)(12,19)(13,17)(14,30)(15,18)(16,22)(20,27)(23,35)(24,34)(25,39)(26,38)(28,37)(29,36)(31,32)(33,40);(1,2,4,6)(3,8,7,5)(9,11,15,25)(10,16,18,33)(12,26,28,23)(13,22,29,40)(14,31,27,24)(17,32,36,34)(19,21,37,39)(20,38,30,35);(10,17)(12,13)(14,27)(16,32)(18,36)(19,30)(20,37)(21,35)(22,26)(23,40)(24,31)(28,29)(33,34)(38,39);(10,18)(16,33)(17,36)(19,37)(20,30)(21,39)(32,34)(35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396254);