#139407

order	512
nilpotency class	4
isoclinism	isoclinic class of #139360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 420 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,d^2=ghi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,d^c=dhi,f^c=fhi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46603);