Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6370897 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#381 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4163 conjugacy classes of subgroups; only the 1296 normal subgroups are shown.
This group has 4163 conjugacy classes of subgroups — too many to draw. Only its 1296 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,13,17,8)(4,18)(5,10)(6,11)(7,12)(9,21)(14,26,40,35)(15,36,42,27)(16,37,44,28)(19,45)(20,46)(22,31)(23,32)(24,33)(25,34)(29,48)(30,50)(38,51,60,57)(39,52,62,58)(41,54,43,53)(47,56)(49,55)(59,63,61,64);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,26)(10,27)(11,28)(12,13)(18,35)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(29,51)(30,52)(31,53)(32,36)(33,54)(34,37)(45,57)(46,58)(47,59)(48,60)(49,61)(50,62)(55,63)(56,64);(1,4)(2,9,12,18)(3,14)(5,19)(6,20)(7,21)(8,26,13,35)(10,29,32,45)(11,30,34,46)(15,38)(16,39)(17,40)(22,47)(23,48)(24,49)(25,50)(27,51,36,57)(28,52,37,58)(31,55,33,56)(41,59)(42,60)(43,61)(44,62)(53,63,54,64);(1,5,7,23)(2,10,12,32)(3,15,17,42)(4,19,21,48)(6,24,25,22)(8,27,13,36)(9,29,18,45)(11,33,34,31)(14,38,40,60)(16,43,44,41)(20,49,50,47)(26,51,35,57)(28,54,37,53)(30,56,46,55)(39,61,62,59)(52,64,58,63);(1,6,7,25)(2,11,12,34)(3,16,17,44)(4,20,21,50)(5,22,23,24)(8,28,13,37)(9,30,18,46)(10,31,32,33)(14,39,40,62)(15,41,42,43)(19,47,48,49)(26,52,35,58)(27,53,36,54)(29,55,45,56)(38,59,60,61)(51,63,57,64)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=i,e^2=i,b^a=bf,c^a=cg,f^a=fi,h^b=hi,g^c=gi,e^d=ei113 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278094);