#131332

order	512
nilpotency class	4
isoclinism	isoclinic class of #131330
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	#60768	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 768 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)(3,4,7,8)(5,6)(9,10,12,19,27,36,39,37,13,18,28,17,14,20,29,31)(11,21,23,38,30,34,26,32,24,16,22,33,25,35,40,15);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,26)(13,24)(14,30)(16,31)(17,35)(18,32)(19,34)(20,38)(21,37)(22,29)(23,39)(25,27)(28,40)(33,36);(1,4)(2,3)(5,8)(6,7)(9,11)(10,16)(12,22)(13,24)(14,25)(15,31)(17,33)(18,21)(19,38)(20,34)(23,28)(26,29)(27,30)(32,37)(35,36)(39,40)

Polycyclic presentation:

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

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, 38528);