Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431846 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#393 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3687 conjugacy classes of subgroups; only the 904 normal subgroups are shown.
This group has 3687 conjugacy classes of subgroups — too many to draw. Only its 904 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)(3,8)(4,17,21,9)(5,10)(6,11)(7,12)(13,36,18,26)(14,27)(15,28)(16,29)(19,43,48,30)(20,44,50,31)(22,32)(23,33)(24,34)(25,35)(37,59,45,51)(38,60,46,52)(39,53)(40,54)(41,55)(42,56)(47,58,49,57)(61,64,62,63);(1,3)(2,8)(4,18,21,13)(5,14)(6,15)(7,16)(9,26,17,36)(10,27)(11,28)(12,29)(19,45,48,37)(20,46,50,38)(22,39)(23,40)(24,41)(25,42)(30,51,43,59)(31,52,44,60)(32,53)(33,54)(34,55)(35,56)(47,62,49,61)(57,63,58,64);(1,4)(2,9)(3,13)(5,19)(6,20)(7,21)(8,26)(10,30)(11,31)(12,17)(14,37)(15,38)(16,18)(22,47)(23,48)(24,49)(25,50)(27,51)(28,52)(29,36)(32,57)(33,43)(34,58)(35,44)(39,61)(40,45)(41,62)(42,46)(53,63)(54,59)(55,64)(56,60);(1,5,7,23)(2,10,12,33)(3,14,16,40)(4,19,21,48)(6,24,25,22)(8,27,29,54)(9,30,17,43)(11,34,35,32)(13,37,18,45)(15,41,42,39)(20,49,50,47)(26,51,36,59)(28,55,56,53)(31,58,44,57)(38,62,46,61)(52,64,60,63);(1,6,7,25)(2,11,12,35)(3,15,16,42)(4,20,21,50)(5,22,23,24)(8,28,29,56)(9,31,17,44)(10,32,33,34)(13,38,18,46)(14,39,40,41)(19,47,48,49)(26,52,36,60)(27,53,54,55)(30,57,43,58)(37,61,45,62)(51,63,59,64)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=i,e^2=i,b^a=bf,c^a=cg,g^a=gi,h^a=hi,c^b=ch,h^b=hi,f^c=fi,e^d=ei89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339043);