#151561

order	512
nilpotency class	5
isoclinism	isoclinic class of #151512
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂×D₁₆	D₄
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

192 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

(2,6)(3,7)(4,8)(9,10,15,21)(11,22,27,54)(12,17,30,20)(13,31,34,52)(14,35,37,43)(16,38,41,24)(18,44,47,33)(19,48,50,29)(23,53,26,60)(25,58,59,71)(28,61,64,65)(32,49,67,72)(36,45,56,70)(39,57,51,55)(40,62,68,66)(42,69,46,63);(1,2,5,3)(4,7,8,6)(9,11,15,27)(10,16,21,41)(12,23,30,26)(13,32,34,67)(14,36,37,56)(17,39,20,51)(18,45,47,70)(19,49,50,72)(22,52,54,31)(24,44,38,33)(25,48,59,29)(28,62,64,66)(35,40,43,68)(42,58,46,71)(53,65,60,61)(55,69,57,63);(1,3,5,2)(4,6,8,7)(9,12,15,30)(10,17,21,20)(11,23,27,26)(13,28,34,64)(14,29,37,48)(16,39,41,51)(18,42,47,46)(19,43,50,35)(22,53,54,60)(24,55,38,57)(25,56,59,36)(31,61,52,65)(32,62,67,66)(33,63,44,69)(40,49,68,72)(45,58,70,71)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gi,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,h^a=hi,d^b=df,e^b=eg,f^b=fh,h^b=hi,d^c=dg,e^c=eg

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58757);