Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512724 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#367 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
409 conjugacy classes of subgroups; only the 272 normal subgroups are shown.
This group has 409 conjugacy classes of subgroups — too many to draw. Only its 272 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 — 4 generators on 64 points:
(2,8,11,27)(4,18)(5,22)(6,23)(9,30)(10,25,35,49)(12,31,38,54)(13,41,39,29)(15,45)(16,48)(17,44)(19,24)(20,34)(21,33)(26,53,40,28)(32,55)(36,56,59,64)(37,62,60,52)(42,58)(43,46)(47,57)(50,63,61,51);(1,2,9,31,5,12,4,8,23,39,18,28,19,26,30,41,7,11,34,54,24,38,21,27,6,13,33,53,22,40,20,29)(3,10,32,56,16,36,15,25,44,60,45,51,46,50,55,62,14,35,57,64,43,59,42,49,17,37,58,63,48,61,47,52);(1,3)(2,10)(4,15)(5,16)(6,17)(7,14)(8,25)(9,32)(11,35)(12,36)(13,37)(18,45)(19,46)(20,47)(21,42)(22,48)(23,44)(24,43)(26,50)(27,49)(28,51)(29,52)(30,55)(31,56)(33,58)(34,57)(38,59)(39,60)(40,61)(41,62)(53,63)(54,64);(3,14)(10,35)(15,42)(16,43)(17,44)(25,49)(32,57)(36,59)(37,60)(45,58)(46,48)(47,55)(50,61)(51,63)(52,62)(56,64)Polycyclic presentation:
2^9|a^2=f,b^2=egh,e^2=ghi,g^2=hi,h^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,f^b=fi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224987, #72225001, #72225002, … |
| maximal normal subgroup of | #72224987, #72225001, #72225002, … |
| maximal quotient of | #72232012, #72232024, #72234330, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419920);