#151306

order	512
nilpotency class	5
isoclinism	isoclinic class of #151260
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	C₂³	B(4,𝔽₂)
LCS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #391 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 279 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)(9,10,15,21)(11,22,27,36)(12,28,24,16)(13,18,23,19)(14,20,32,39)(17,26,37,29)(25,34,40,35)(30,33,31,38);(1,2,4,6)(3,8,7,5)(9,11)(10,16)(12,23)(13,24)(14,26)(15,27)(17,33)(18,22)(19,36)(20,34)(21,28)(25,31)(29,32)(30,40)(35,39)(37,38);(9,12,14,25,15,24,32,40)(10,17,20,36,21,37,39,22)(11,23,26,31,27,13,29,30)(16,33,34,19,28,38,35,18)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,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=ehi,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, 58502);