#145861

order	512
nilpotency class	4
isoclinism	isoclinic class of #143319
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1629
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#453
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

291 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 84 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)(9,10,12,16)(11,19,21,15)(13,27,25,40)(14,29,26,18)(17,35,34,28)(20,32,38,37)(22,31,24,33)(23,30,39,36);(1,3,5,4)(2,6,7,8)(9,11,12,21)(10,15,16,19)(13,20,25,38)(14,22,26,24)(17,30,34,36)(18,31,29,33)(23,28,39,35)(27,37,40,32);(1,4,5,3)(2,6,7,8)(11,20,21,38)(13,28)(14,26)(15,30,19,36)(17,27)(18,29)(22,23,24,39)(25,35)(31,32,33,37)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226865, #72226872, #72226873
maximal normal subgroup of	#72226865, #72226872, #72226873

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53057);