#138694

order	512
nilpotency class	4
isoclinism	isoclinic class of #138694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

333 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 333 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=ehi,f^c=fhi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45890);