#146068

order	512
nilpotency class	4
isoclinism	isoclinic class of #146048
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2073
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 238 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:

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

Polycyclic presentation:

2^9|a^2=egh,c^2=d,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fh,h^b=hi,e^c=eg,f^c=fh,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, 53264);