Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1924 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
735 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,7,14,30,46,8,11,31,43,5,12,16,33)(3,15,20,47,18,35,48,44,21,36,28,59,19,9,32,37)(4,22,26,55,25,40,49,45,27,41,29,60,24,10,38,42)(17,50,51,62,23,56,57,64,52,34,54,61,53,39,58,63);(1,3,16,48,5,19,31,20,8,21,30,32,7,18,6,28)(2,9,33,47,12,36,43,37,11,35,46,59,14,15,13,44)(4,23,38,54,24,17,29,57,27,53,49,51,25,52,26,58)(10,39,60,62,41,34,45,63,40,56,55,61,22,50,42,64);(1,4,5,24,8,27,7,25)(2,10,12,41,11,40,14,22)(3,17,18,23,21,52,19,53)(6,29,16,49,31,26,30,38)(9,34,15,39,35,50,36,56)(13,45,33,55,43,42,46,60)(20,54,48,58,28,51,32,57)(37,62,47,64,44,61,59,63);(2,11)(9,35)(10,40)(12,14)(13,43)(15,36)(22,41)(33,46)(34,50)(37,44)(39,56)(42,45)(47,59)(55,60)(61,62)(63,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ch,e^b=ei,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402036);