Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6489185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #35237 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3584 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1691 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,10,8,14,6,13)(3,15,16,9,20,32,19,33)(4,11,24,34,25,39,23,38)(5,26,28,40,29,35,27,12)(17,36,42,30,45,57,21,49)(18,46,47,55,48,31,43,41)(22,51,53,60,54,58,52,37)(44,62,50,64,63,56,61,59);(1,3)(2,9)(4,21)(5,18)(6,16)(7,19)(8,20)(10,15)(11,36)(12,31)(13,32)(14,33)(17,23)(22,50)(24,45)(25,42)(26,55)(27,47)(28,43)(29,48)(30,38)(34,49)(35,41)(37,59)(39,57)(40,46)(44,53)(51,56)(52,63)(54,61)(58,62)(60,64);(2,10,14,13)(3,16,20,19)(11,34,39,38)(12,35,40,26)(17,42,45,21)(18,43,48,47)(37,58,60,51)(44,61,63,50);(1,4,7,24,8,25,6,23)(2,11,10,34,14,39,13,38)(3,17,16,42,20,45,19,21)(5,22,27,52,29,54,28,53)(9,30,32,57,33,49,15,36)(12,37,35,58,40,60,26,51)(18,44,43,61,48,63,47,50)(31,56,55,64,46,62,41,59);(1,5,7,27,8,29,6,28)(2,12,13,26,14,40,10,35)(3,18,19,47,20,48,16,43)(4,22,24,52,25,54,23,53)(9,31,32,55,33,46,15,41)(11,37,38,51,39,60,34,58)(17,44,21,50,45,63,42,61)(30,56,57,64,49,62,36,59)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,d^2=hi,e^2=g,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396384);