#489063

order	512
nilpotency class	4
isoclinism	isoclinic class of #488920
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	C₂²×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

718 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 156 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)(9,10,15,20)(11,21,22,40)(12,18,28,38)(13,17,29,37)(14,30,27,19)(16,32,34,23)(24,36,26,39)(25,35,31,33);(1,2,4,6)(3,8,7,5)(9,11,15,22)(10,16,20,34)(12,26,28,24)(13,23,29,32)(14,31,27,25)(17,33,37,35)(18,21,38,40)(19,39,30,36);(10,17)(12,13)(14,27)(16,33)(18,30)(19,38)(20,37)(21,36)(23,26)(24,32)(25,31)(28,29)(34,35)(39,40);(11,22)(16,34)(21,40)(23,32)(24,26)(25,31)(33,35)(36,39)

Polycyclic presentation:

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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396259);