#151547

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×D₄	D₁₆
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 190 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,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, 58743);