#503530

order	512
nilpotency class	4
isoclinism	isoclinic class of #501076
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×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

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

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,d^c=di,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410726);