Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500740 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24236 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1007 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,9,16)(3,17,23,41)(4,24,30,50)(5,15,7,34)(6,32,31,13)(8,12,33,14)(10,22,42,20)(11,37,47,28)(18,43,54,64)(19,48,35,39)(21,40,55,60)(25,38,59,62)(26,46,36,45)(27,49,29,44)(51,58,57,52)(53,61,56,63);(1,3,6,20,9,23,31,22)(2,10,13,17,16,42,32,41)(4,25,27,58,30,59,29,52)(5,19,8,21,7,35,33,55)(11,43,45,63,47,64,46,61)(12,39,15,40,14,48,34,60)(18,37,53,36,54,28,56,26)(24,57,44,38,50,51,49,62);(1,4)(2,11)(3,18)(5,26)(6,27)(7,36)(8,37)(9,30)(10,38)(12,44)(13,45)(14,49)(15,50)(16,47)(17,51)(19,52)(20,53)(21,25)(22,56)(23,54)(24,34)(28,33)(29,31)(32,46)(35,58)(39,61)(40,43)(41,57)(42,62)(48,63)(55,59)(60,64);(1,5,31,33,9,7,6,8)(2,12,32,34,16,14,13,15)(3,19,22,55,23,35,20,21)(4,26,29,28,30,36,27,37)(10,39,41,60,42,48,17,40)(11,44,46,24,47,49,45,50)(18,52,56,59,54,58,53,25)(38,61,57,64,62,63,51,43)Polycyclic presentation:
2^9|a^2=i,b^2=e,d^2=ei,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi,f^c=fh,g^c=ghi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407938);