#151233

order	512
nilpotency class	5
isoclinism	isoclinic class of #151146
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2196
commutator, Frattini, cosocle	#391	C₂³
prime radical	#2733	C₂²
LCS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #391 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

330 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 38 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fh,d^2=f,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,g^a=gh,d^b=df,e^b=eghi,g^b=gi,d^c=dg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=eh,g^e=gi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58429);