#151226

order	512
nilpotency class	5
isoclinism	isoclinic class of #151146
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	#537	D₄
LCS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #391 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

309 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 38 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,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, 58422);