#145364

order	512
nilpotency class	4
isoclinism	isoclinic class of #145363
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	#3147	C₂²
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

314 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=d,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=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fgi,g^b=gi,h^b=hi,e^c=egh

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, 52560);