#146015

order	512
nilpotency class	4
isoclinism	isoclinic class of #143951
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1667
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

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

normal subgroups only

This group has 237 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53211);