Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #434061 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15496 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1127 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,6,12,8,14)(3,15,29,34,18,9,19,40)(4,20,23,37,24,41,30,10)(5,25,31,39,27,11,28,42)(16,35,21,43,46,58,45,32)(17,33,47,59,26,38,48,60)(22,51,53,55,54,49,52,36)(44,63,64,62,56,61,50,57);(1,3,6,18)(2,9,12,15)(4,21)(5,26,27,17)(7,29,8,19)(10,35)(11,38,25,33)(13,40,14,34)(16,30)(20,43)(22,50)(23,46)(24,45)(28,47,31,48)(32,41)(36,61)(37,58)(39,59,42,60)(44,53)(49,62)(51,57)(52,56)(54,64)(55,63);(1,4,8,23,6,24,7,30)(2,10,14,20,12,37,13,41)(3,16,29,45,18,46,19,21)(5,22,31,52,27,54,28,53)(9,32,40,58,15,43,34,35)(11,36,42,49,25,55,39,51)(17,44,48,64,26,56,47,50)(33,57,60,63,38,62,59,61);(1,5,6,27)(2,11,12,25)(3,17,18,26)(4,22,24,54)(7,28,8,31)(9,33,15,38)(10,36,37,55)(13,39,14,42)(16,44,46,56)(19,47,29,48)(20,49,41,51)(21,50,45,64)(23,52,30,53)(32,57,43,62)(34,59,40,60)(35,61,58,63)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=ef,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341258);