#131930

order	512
nilpotency class	4
isoclinism	isoclinic class of #131808
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#3355	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

612 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=hi,e^2=g,f^2=gh,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=eg,f^c=fi

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, 39126);