#512109

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

957 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)(3,6)(4,5)(7,8)(9,10,16,23)(11,17,28,46)(12,29,33,51)(13,18,37,21)(14,38,34,20)(15,39,36,22)(19,35,50,31)(24,53,57,70)(25,41,61,44)(26,62,58,43)(27,63,60,45)(30,49,65,52)(32,48,40,47)(42,59,69,55)(54,68,72,71)(56,67,64,66);(2,5)(6,8)(9,11,16,28)(10,17,23,46)(12,24,33,57)(13,25,37,61)(14,26,34,58)(15,27,36,60)(18,41,21,44)(19,42,50,69)(20,43,38,62)(22,45,39,63)(29,53,51,70)(30,54,65,72)(31,55,35,59)(32,56,40,64)(47,66,48,67)(49,68,52,71);(1,3,4,7)(2,6,5,8)(10,18)(13,34)(14,37)(15,36)(17,41)(19,47)(20,22)(21,23)(25,58)(26,61)(27,60)(29,49)(30,31)(32,40)(35,65)(38,39)(42,66)(43,45)(44,46)(48,50)(51,52)(53,68)(54,55)(56,64)(59,72)(62,63)(67,69)(70,71);(9,12)(10,19)(11,24)(13,35)(14,30)(15,40)(16,33)(17,42)(18,47)(20,51)(21,48)(22,52)(23,50)(25,59)(26,54)(27,64)(28,57)(29,38)(31,37)(32,36)(34,65)(39,49)(41,66)(43,70)(44,67)(45,71)(46,69)(53,62)(55,61)(56,60)(58,72)(63,68)

Polycyclic presentation:

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

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, 419305);