Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420401 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15277 |
| 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 | |
1289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,31)(3,14,17,34,19,42,18,9)(4,20,24,38,25,37,23,10)(5,26,29,40,30,11,28,41)(15,32,21,51,45,49,44,35)(16,46,47,59,48,39,27,33)(22,50,53,36,55,62,54,56)(43,58,57,60,63,61,52,64);(1,3,8,19)(2,9,13,34)(4,21,25,44)(5,27,30,47)(6,18,7,17)(10,35,38,51)(11,39,26,46)(12,42,31,14)(15,23,45,24)(16,29,48,28)(20,49,37,32)(22,43,55,63)(33,41,59,40)(36,61,56,58)(50,64,62,60)(52,54,57,53);(1,4,6,23,8,25,7,24)(2,10,12,37,13,38,31,20)(3,15,17,44,19,45,18,21)(5,22,28,53,30,55,29,54)(9,32,14,35,34,49,42,51)(11,36,40,62,26,56,41,50)(16,43,27,57,48,63,47,52)(33,58,39,60,59,61,46,64);(1,5,6,28,8,30,7,29)(2,11,12,40,13,26,31,41)(3,16,17,27,19,48,18,47)(4,22,23,53,25,55,24,54)(9,33,14,39,34,59,42,46)(10,36,37,62,38,56,20,50)(15,43,44,57,45,63,21,52)(32,58,35,60,49,61,51,64)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=e,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327598);