#146074

order	512
nilpotency class	4
isoclinism	isoclinic class of #144059
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1677
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 223 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)(3,6)(4,5)(7,8)(9,10,12,19,15,22,29,20,16,23,30,37,14,21,13,18)(11,17,26,38,31,32,40,39,28,35,25,33,27,34,24,36);(2,5)(6,8)(9,11)(10,17)(12,26)(13,24)(14,27)(15,31)(16,28)(18,36)(19,38)(20,39)(21,34)(22,32)(23,35)(25,30)(29,40)(33,37);(1,3,4,7)(2,6,5,8)(10,18)(11,24)(12,13)(14,15)(17,32)(19,21)(20,23)(22,37)(25,31)(26,27)(28,40)(29,30)(34,35)(36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53270);