#145802

order	512
nilpotency class	4
isoclinism	isoclinic class of #142406
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

324 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52998);