#131916

order	512
nilpotency class	4
isoclinism	isoclinic class of #131796
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	#532	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 562 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) = 256

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

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

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=eg,f^b=fgh,h^b=hi,d^c=dg,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, 39112);