#145449

order	512
nilpotency class	4
isoclinism	isoclinic class of #145359
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#463
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 276 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:

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

Polycyclic presentation:

2^9|a^2=gh,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=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fg,g^b=gi,h^b=hi,e^c=egh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52645);