Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12711 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1781 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | #10725 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1605 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,15)(4,21,24,38)(5,25,27,11)(6,12)(7,13)(8,14)(9,17,32,48)(10,23,35,29)(16,43,46,62)(18,40)(19,41)(20,42)(22,53)(26,37,28,36)(30,58)(31,47,33,49)(34,54)(39,45,61,50)(44,60)(51,63)(52,55)(56,57)(59,64);(1,3,7,19,8,20,6,18)(2,9,13,33,14,32,12,31)(4,16,23,45,24,46,29,50)(5,17,28,49,27,48,26,47)(10,30,21,51,35,59,38,60)(11,15,37,41,25,42,36,40)(22,44,54,58,55,63,57,64)(34,39,52,43,56,61,53,62);(1,4)(2,10)(3,16)(5,22)(6,29)(7,23)(8,24)(9,30)(11,34)(12,38)(13,21)(14,35)(15,39)(17,44)(18,50)(19,45)(20,46)(25,56)(26,57)(27,55)(28,54)(31,60)(32,59)(33,51)(36,53)(37,52)(40,62)(41,43)(42,61)(47,64)(48,63)(49,58)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,f^b=fh,e^c=ei,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72317336, #72317352, #72317375, … |
| maximal normal subgroup of | #72317336, #72317352, #72317375, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53682);