#151149

order	512
nilpotency class	5
isoclinism	isoclinic class of #151146
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

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

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

normal subgroups only

This group has 567 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)(9,10,15,16)(11,20,23,34)(12,25,27,29)(13,14,30,19)(17,28,24,22)(18,31,32,26)(21,33,37,39)(35,38,40,36);(2,5)(3,8)(6,7)(9,11)(10,17)(12,21)(13,28)(14,20)(15,23)(16,24)(18,33)(19,34)(22,30)(25,38)(26,40)(27,37)(29,36)(31,35)(32,39);(9,12,15,27)(10,18,16,32)(11,21,23,37)(13,29,30,25)(14,31,19,26)(17,33,24,39)(20,35,34,40)(22,38,28,36)

Polycyclic presentation:

2^9|a^2=i,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.

Relationships

maximal quotient of

#72267844, #72267850, #72267851

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58345);