Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #147593 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10965 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,7,12,8,13,6,26)(3,14,18,45,17,43,27,44)(4,10,21,37,22,38,28,36)(5,23,24,39,25,11,29,40)(9,30,33,16,32,51,41,50)(15,46,49,63,48,42,19,52)(20,54,56,62,57,35,55,61)(31,53,60,64,59,58,34,47);(1,3)(2,9)(4,19)(5,16)(6,27)(7,18)(8,17)(10,34)(11,14)(12,41)(13,32)(15,21)(20,53)(22,49)(23,43)(24,51)(25,50)(26,33)(28,48)(29,30)(31,36)(35,52)(37,59)(38,60)(39,45)(40,44)(42,61)(46,62)(47,55)(54,63)(56,64)(57,58);(1,4,8,22)(2,10,13,38)(3,15,17,48)(5,20,25,57)(6,28,7,21)(9,31,32,59)(11,35,23,54)(12,37,26,36)(14,42,43,46)(16,47,50,64)(18,49,27,19)(24,56,29,55)(30,58,51,53)(33,60,41,34)(39,62,40,61)(44,63,45,52)Polycyclic presentation:
2^9|a^2=g,c^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gh,f^c=fi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72267972, #72267974, #72267977 |
| maximal normal subgroup of | #72267972, #72267974, #72267977 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54792);