#143068

order	512
nilpotency class	4
isoclinism	isoclinic class of #143006
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂²×D₄	#41
LCS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

603 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 603 conjugacy classes of subgroups — too many to draw. Only its 58 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)(9,10,15,20)(11,21,25,32)(12,26,30,35)(13,19,14,18)(16,24,33,23)(17,28,36,29)(22,31,38,39)(27,40,37,34);(1,2)(3,5)(4,6)(7,8)(9,11,15,25)(10,16,20,33)(12,27,30,37)(13,24,14,23)(17,34,36,40)(18,32,19,21)(22,29,38,28)(26,39,35,31);(3,7)(9,12,15,30)(10,17,20,36)(11,22,25,38)(13,28,14,29)(16,31,33,39)(18,35,19,26)(21,34,32,40)(23,27,24,37)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fg,h^b=hi,d^c=dh,e^c=eg,f^c=fg,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50264);