#151572

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

230 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^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, 58768);