#131372

order	512
nilpotency class	4
isoclinism	isoclinic class of #131284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#3259	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

675 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 675 conjugacy classes of subgroups — too many to draw. Only its 98 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)(3,4,8,6)(5,7)(9,10,12,18,13,29,27,33,15,20,28,32,14,19,26,16)(11,21,24,39,30,38,40,35,25,37,22,31,23,17,34,36);(1,3)(2,6)(4,7)(5,8)(10,16)(12,26)(13,14)(17,31)(18,19)(20,33)(21,35)(22,40)(23,30)(24,34)(27,28)(29,32)(36,37)(38,39);(1,4,5,6)(2,3,7,8)(9,11,15,25)(10,17,20,38)(12,24,28,22)(13,30,14,23)(16,31,33,39)(18,36,32,35)(19,37,29,21)(26,34,27,40)

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,e^c=ei,f^c=fg

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38568);