#146267

order	512
nilpotency class	4
isoclinism	isoclinic class of #146221
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#496
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

230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=dh,b^2=deg,c^2=g,d^2=gi,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,d^c=dg,e^c=eg,f^c=fh,h^c=hi

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, 53463);