#138913

order	512
nilpotency class	4
isoclinism	isoclinic class of #138832
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#2186	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 540 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,d^2=ghi,e^2=ghi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=egh,f^b=fh,h^b=hi,d^c=dgh,e^c=egh,f^c=fgh,h^c=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46109);