Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512698 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#357 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| prime radical | #62635 | C2 |
| 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 | |
763 conjugacy classes of subgroups; only the 272 normal subgroups are shown.
This group has 763 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:
(2,8,10,30)(4,17)(5,22)(6,24)(9,28,33,54)(11,37,36,51)(12,38,26,32)(14,42)(15,46)(16,41)(18,23,31,29)(19,25)(20,52)(21,27)(34,59,58,63)(35,60,48,57)(39,53)(40,44)(43,47,56,55)(45,64)(49,50)(61,62);(1,2)(3,9)(4,18)(5,23)(6,26)(7,10)(8,27)(11,22)(12,24)(13,33)(14,43)(15,47)(16,48)(17,30)(19,36)(20,37)(21,31)(25,29)(28,53)(32,50)(34,46)(35,41)(38,52)(39,56)(40,55)(42,54)(44,58)(45,59)(49,51)(57,62)(60,64)(61,63);(1,3)(2,9)(4,14)(5,15)(6,16)(7,13)(8,28)(10,33)(11,34)(12,35)(17,42)(18,43)(19,44)(20,45)(21,39)(22,46)(23,47)(24,41)(25,40)(26,48)(27,53)(29,55)(30,54)(31,56)(32,57)(36,58)(37,59)(38,60)(49,61)(50,62)(51,63)(52,64);(3,13)(9,33)(14,39)(15,40)(16,41)(28,54)(34,58)(35,48)(42,53)(43,56)(44,46)(45,61)(47,55)(57,60)(59,63)(62,64)Polycyclic presentation:
2^9|a^2=f,e^2=ghi,g^2=hi,h^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,e^b=egi,f^b=fi,g^b=gh,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224987, #72225013, #72225014, … |
| maximal normal subgroup of | #72224987, #72225013, #72225014, … |
| maximal quotient of | #72232008, #72232018, #72234322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419894);