#151165

order	512
nilpotency class	5
isoclinism	isoclinic class of #151162
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	#382	C₂³
prime radical	#31075	C₂
LCS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #382 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 399 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)(3,5)(4,7)(6,8)(9,10,15,12)(11,19,23,37)(13,17,27,24)(14,25,30,18)(16,31,20,21)(22,36,40,35)(26,29,33,28)(32,39,38,34);(2,5)(3,8)(6,7)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,19)(18,32)(24,37)(25,38)(26,35)(27,31)(28,34)(29,39)(30,40)(33,36);(9,12,15,10)(11,20,23,16)(13,26,27,33)(14,18,30,25)(17,28,24,29)(19,34,37,39)(21,35,31,36)(22,32,40,38)

Polycyclic presentation:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72267844

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58361);