#489059

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	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 706 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=gh,c^2=i,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, 396255);