#506462

order	512
nilpotency class	4
isoclinism	isoclinic class of #500681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 160 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 — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egi,c^2=i,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,g^c=gi,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, 413658);