#489068

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
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 ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 691 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)(11,22)(12,25)(13,31)(14,21)(15,20)(16,18)(17,24)(19,34)(23,28)(26,40)(27,33)(29,32)(30,38)(35,37)(36,39);(1,2,4,6)(3,8,7,5)(9,11)(10,17)(12,18)(13,32)(14,27)(15,26)(16,23)(19,33)(20,29)(21,30)(22,31)(24,34)(25,35)(28,39)(36,38)(37,40);(9,12,16,17)(10,11,18,23)(13,33,36,40)(14,30,15,29)(19,38,37,32)(20,27,21,26)(22,35,28,34)(24,31,25,39);(10,18)(11,23)(19,37)(20,21)(24,25)(26,27)(31,39)(32,38)

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,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396264);