#151303

order	512
nilpotency class	5
isoclinism	isoclinic class of #151256
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	#390	C₂³
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #390 ◁ G
LECS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

292 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 40 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)(3,7)(6,8)(9,10,14,18)(11,15,21,29)(12,16,25,32)(13,26,22,17)(19,33,37,39)(20,28,34,38)(23,30,24,31)(27,36,40,35);(1,2,4,6)(3,8,7,5)(9,11)(10,15)(12,19)(13,20)(14,21)(16,27)(17,28)(18,29)(22,34)(23,35)(24,36)(25,37)(26,38)(30,33)(31,39)(32,40);(9,12,22,23,14,25,13,24)(10,16,26,30,18,32,17,31)(11,19,34,35,21,37,20,36)(15,27,38,33,29,40,28,39)

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58499);