#489200

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	#33716	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

767 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 767 conjugacy classes of subgroups — too many to draw. Only its 146 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)(11,20)(12,15)(13,18)(14,28)(16,25)(17,23)(19,31)(21,37)(22,38)(24,39)(26,27)(29,33)(30,36)(32,35)(34,40);(3,7)(5,8)(9,11,15,24)(10,17,12,21)(13,22,16,32)(14,29,30,40)(18,33,25,34)(19,35,26,38)(20,36,39,28)(23,27,37,31);(1,2,4,6)(3,8,7,5)(9,12,15,10)(11,21,24,17)(13,18,16,25)(14,19,30,26)(20,37,39,23)(22,33,32,34)(27,36,31,28)(29,35,40,38);(9,13,15,16)(10,18,12,25)(11,22,24,32)(14,27,30,31)(17,33,21,34)(19,28,26,36)(20,38,39,35)(23,29,37,40)

Polycyclic presentation:

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

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, 396396);