#503527

order	512
nilpotency class	4
isoclinism	isoclinic class of #501072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

771 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,c^2=ef,d^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410723);