#146073

order	512
nilpotency class	4
isoclinism	isoclinic class of #146043
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2077
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

225 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=egh,c^2=d,d^2=i,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=di,f^b=fh,e^c=eg,f^c=fhi,g^c=gi

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, 53269);