#144513

order	512
nilpotency class	4
isoclinism	isoclinic class of #143203
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1619
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

329 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 80 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)(12,24)(13,28)(15,30)(16,19)(17,33)(21,39)(22,29)(25,36)(26,40)(27,34)(31,32)(35,38);(1,3,4,8)(2,5,6,7)(9,10)(11,20)(12,25)(13,16)(14,18)(15,21)(17,34)(19,28)(22,35)(23,37)(24,36)(26,32)(27,33)(29,38)(30,39)(31,40);(9,11,14,23)(10,15,18,31)(12,26,27,39)(13,29,28,22)(16,32,19,30)(17,35,36,38)(20,33,37,25)(21,24,40,34)

Polycyclic presentation:

2^9|a^2=d,b^2=d,c^2=i,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=fhi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226868, #72226869, #72226872
maximal normal subgroup of	#72226868, #72226869, #72226872

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51709);