#141186

order	512
nilpotency class	4
isoclinism	isoclinic class of #140953
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#33179	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

470 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=e,d^2=ghi,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fhi,g^b=gi,h^b=hi,d^c=di,f^d=fi

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, 48382);