#130609

order	512
nilpotency class	4
isoclinism	isoclinic class of #130518
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2353 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

507 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=i,d^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ei,f^b=fghi,d^c=di,f^c=fg,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37805);