Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351817 |
| 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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2588 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)(3,13)(4,17)(5,11)(6,12)(7,10)(8,15)(9,19)(14,22)(16,24)(18,25)(20,23)(21,26)(27,30)(28,32)(29,31)(33,34)(35,44)(36,41)(37,42)(38,51)(39,43)(40,47)(45,60)(46,54)(48,58)(49,55)(50,56)(52,57)(53,62)(59,61)(63,64);(1,3)(2,8)(4,14)(5,20)(6,15)(7,16)(9,22)(10,23)(11,13)(12,24)(17,31)(18,28)(19,30)(21,29)(25,27)(26,32)(33,35,39,48)(34,40,43,54)(36,45,49,61)(37,47,38,46)(41,53,55,64)(42,58,51,44)(50,63,52,62)(56,60,57,59);(1,4)(2,9)(3,14)(5,19)(6,21)(7,18)(8,22)(10,26)(11,25)(12,17)(13,27)(15,29)(16,28)(20,30)(23,32)(24,31)(33,36)(34,41)(35,45)(37,50)(38,52)(39,49)(40,53)(42,56)(43,55)(44,59)(46,62)(47,63)(48,61)(51,57)(54,64)(58,60);(2,10)(4,18)(8,23)(11,12)(13,24)(14,28)(19,21)(29,30)(36,49)(41,55)(45,61)(50,52)(53,64)(56,57)(59,60)(62,63);(1,5,7,6)(2,11,10,12)(3,15,16,20)(4,19,18,21)(8,24,23,13)(9,25,26,17)(14,29,28,30)(22,31,32,27)(33,37,39,38)(34,42,43,51)(35,46,48,47)(36,50,49,52)(40,44,54,58)(41,56,55,57)(45,62,61,63)(53,59,64,60)Polycyclic presentation:
2^9|b^2=i,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,g^a=ghi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259028);