#145394

order	512
nilpotency class	4
isoclinism	isoclinic class of #145359
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#463
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

308 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,5)(4,6)(7,8)(9,10,16,22)(11,23,29,43)(12,30,35,67)(13,20,37,54)(14,19,39,52)(15,40,36,21)(17,26,46,25)(18,47,51,32)(24,57,61,65)(27,55,62,45)(28,64,41,44)(31,48,63,72)(33,50,38,66)(34,49,69,53)(42,68,71,60)(56,58,70,59);(2,5)(6,7)(9,11,16,29)(10,17,22,46)(12,31,35,63)(13,27,37,62)(14,25,39,26)(15,41,36,28)(18,48,51,72)(19,44,52,64)(20,23,54,43)(21,55,40,45)(24,38,61,33)(30,65,67,57)(32,68,47,60)(34,59,69,58)(42,53,71,49)(50,70,66,56);(2,6)(4,8)(5,7)(9,12,14,38,15,34,37,47,16,35,39,33,36,69,13,32)(10,18,20,53,21,50,52,67,22,51,54,49,40,66,19,30)(11,24,27,63,28,60,26,59,29,61,62,31,41,68,25,58)(17,42,44,72,45,57,43,70,46,71,64,48,55,65,23,56)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=dh,d^2=gi,e^2=hi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=egi,f^b=fgi,g^b=gi,h^b=hi,e^c=eghi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52590);