#145366

order	512
nilpotency class	4
isoclinism	isoclinic class of #145365
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 325 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) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=d,d^2=gi,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,e^c=egh,f^c=fi

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, 52562);