Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351885 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,15)(4,17)(5,11)(6,12)(7,13)(8,14)(9,18)(10,20)(16,29)(19,31)(21,26)(22,27)(23,28)(24,30)(25,32)(34,39)(36,46)(37,48)(40,53)(41,56)(42,49)(43,58)(44,50)(51,55)(52,57)(54,62)(63,64);(1,3,8,10)(2,9,14,4)(5,16,23,25)(6,17,7,18)(11,24,28,19)(12,15,13,20)(21,30,22,31)(26,32,27,29)(33,34,38,42)(35,40,45,55)(36,41,46,57)(37,49,48,39)(43,54,58,64)(44,51,50,53)(47,56,60,52)(59,62,61,63);(1,4,8,9)(2,10,14,3)(5,19,23,24)(6,20,7,15)(11,25,28,16)(12,18,13,17)(21,29,22,32)(26,31,27,30)(33,35,38,45)(34,40,42,55)(36,43,46,58)(37,50,48,44)(39,51,49,53)(41,54,57,64)(47,61,60,59)(52,62,56,63);(1,5,8,23)(2,11,14,28)(3,16,10,25)(4,19,9,24)(6,22,7,21)(12,27,13,26)(15,29,20,32)(17,31,18,30)(33,36,38,46)(34,41,42,57)(35,43,45,58)(37,47,48,60)(39,52,49,56)(40,54,55,64)(44,59,50,61)(51,62,53,63);(1,6,8,7)(2,12,14,13)(3,17,10,18)(4,20,9,15)(5,21,23,22)(11,26,28,27)(16,30,25,31)(19,29,24,32)(36,46)(41,57)(43,58)(47,60)(52,56)(54,64)(59,61)(62,63)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=h,e^2=hi,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=ghi,f^b=fh,g^b=gi,f^c=fi,g^c=ghi,e^d=eh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259223);