#151376

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2196
commutator, Frattini, cosocle	#391	C₂³
prime radical	#23891	C₂
LCS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #391 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 526 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,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,h^b=hi,d^c=dg,e^c=ei,g^c=gi,h^c=hi,e^d=ehi,g^d=gi,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, 58572);