#151565

order	512
nilpotency class	5
isoclinism	isoclinic class of #151520
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	#2184
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

183 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 183 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58761);