Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6355203 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #3511 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,10)(3,11,13,24)(4,8,16,22)(5,18,12,9)(7,19,20,15)(14,25,27,32)(17,29,26,23)(21,30,31,28)(33,34,38,42)(35,39,44,53)(36,46,48,56)(37,49,51,41)(40,50,55,47)(43,57,58,64)(45,61,60,54)(52,62,63,59);(1,3,12,19,6,13,5,15)(2,7,18,24,10,20,9,11)(4,14,26,30,16,27,17,28)(8,21,29,32,22,31,23,25)(33,35)(34,39)(36,43)(37,45)(38,44)(40,52)(41,54)(42,53)(46,57)(47,59)(48,58)(49,61)(50,62)(51,60)(55,63)(56,64);(3,13)(7,20)(11,24)(14,27)(15,19)(21,31)(25,32)(28,30)(33,36,38,48)(34,40,42,55)(35,43,44,58)(37,50,51,47)(39,52,53,63)(41,56,49,46)(45,62,60,59)(54,64,61,57);(1,4)(2,8)(3,14)(5,17)(6,16)(7,21)(9,23)(10,22)(11,25)(12,26)(13,27)(15,28)(18,29)(19,30)(20,31)(24,32)(35,44)(39,53)(43,58)(45,60)(52,63)(54,61)(57,64)(59,62);(4,16)(8,22)(14,27)(17,26)(21,31)(23,29)(25,32)(28,30)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6262426);