#146071

order	512
nilpotency class	4
isoclinism	isoclinic class of #146048
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2073
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

222 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=i,c^2=d,d^2=i,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,h^b=hi,e^c=eg,f^c=fhi,h^c=hi

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, 53267);