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 | |
2587 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)(4,21)(5,11)(6,12)(7,13)(8,14)(9,19)(10,22)(16,26)(17,29)(18,28)(20,32)(23,30)(24,31)(25,27)(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)(2,9)(4,16)(5,17)(6,18)(7,19)(8,20)(10,26)(11,27)(12,15)(13,32)(14,28)(21,30)(22,24)(23,25)(29,31)(33,34)(35,40)(36,41)(37,49)(38,42)(39,48)(43,54)(44,51)(45,55)(46,57)(47,56)(50,53)(52,60)(58,64)(59,62)(61,63);(1,4,8,17)(2,10,14,27)(3,16,20,5)(6,22,7,25)(9,26,28,11)(12,29,13,21)(15,31,32,30)(18,24,19,23)(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);(1,5,8,16)(2,11,14,26)(3,17,20,4)(6,24,7,23)(9,27,28,10)(12,31,13,30)(15,29,32,21)(18,22,19,25)(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)(3,18,20,19)(4,22,17,25)(5,23,16,24)(9,15,28,32)(10,29,27,21)(11,30,26,31)(36,46)(41,57)(43,58)(47,60)(52,56)(54,64)(59,61)(62,63)Polycyclic presentation:
2^9|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, 6259221);