#489037

order	512
nilpotency class	4
isoclinism	isoclinic class of #488889
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	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

728 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 160 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,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, 396233);