Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357427 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #60972 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,10)(3,11,13,25)(4,8,15,22)(5,17,18,9)(7,19,21,14)(12,24,26,32)(16,29,28,23)(20,30,31,27)(33,34)(35,40)(36,48)(37,42)(38,52)(39,43)(41,50)(44,57)(45,54)(46,55)(47,61)(49,56)(51,62)(53,60)(58,64)(59,63);(1,3,5,14,6,13,18,19)(2,7,9,11,10,21,17,25)(4,12,16,27,15,26,28,30)(8,20,23,24,22,31,29,32)(33,35)(34,40)(36,44)(37,46)(38,47)(39,45)(41,53)(42,55)(43,54)(48,57)(49,59)(50,60)(51,58)(52,61)(56,63)(62,64);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,23)(10,22)(11,24)(13,26)(14,27)(17,29)(18,28)(19,30)(21,31)(25,32)(33,36,38,50,39,51,37,49)(34,41,42,48,43,56,52,62)(35,44,47,60,45,58,46,59)(40,53,55,57,54,63,61,64);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(24,32)(27,30)(35,45)(40,54)(44,58)(46,47)(53,63)(55,61)(57,64)(59,60);(3,13)(7,21)(11,25)(12,26)(14,19)(20,31)(24,32)(27,30)(33,37,39,38)(34,42,43,52)(35,46,45,47)(36,49,51,50)(40,55,54,61)(41,48,56,62)(44,59,58,60)(53,57,63,64)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^b=eh,d^c=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264648);