#511997

order	512
nilpotency class	4
isoclinism	isoclinic class of #511755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#578	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

924 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:

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

Polycyclic presentation:

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

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, 419193);