#144462

order	512
nilpotency class	4
isoclinism	isoclinic class of #142310
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

508 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=d,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=egi,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, 51658);