#139211

order	512
nilpotency class	4
isoclinism	isoclinic class of #139152
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	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	SD₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

332 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 256

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

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

Polycyclic presentation:

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

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, 46407);