#145537

order	512
nilpotency class	4
isoclinism	isoclinic class of #143938
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2054
prime radical	#33716	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

348 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,5)(4,7)(6,8)(9,10)(11,12)(13,27)(14,34)(15,23)(16,22)(17,24)(18,28)(19,33)(20,36)(21,32)(25,40)(26,31)(29,35)(30,37)(38,39);(1,2)(3,5)(4,7)(6,8)(9,11)(10,17)(12,27)(13,18)(14,35)(15,37)(16,26)(19,30)(20,40)(21,28)(22,38)(23,29)(24,33)(25,32)(31,34)(36,39);(2,5)(3,8)(6,7)(9,12,13,28,32,40,36,38,16,31,14,29,15,30,33,17)(10,18,19,11,23,24,34,37,22,35,20,26,21,39,27,25)

Polycyclic presentation:

2^9|c^2=d,d^2=gi,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ehi,f^b=fh,g^b=gi,h^b=hi,e^c=egh,f^c=fgh

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, 52733);