#146047

order	512
nilpotency class	4
isoclinism	isoclinic class of #146043
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2077
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#468
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

206 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=h,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, 53243);