Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #437530 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15602 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,27,6,12,8,13)(3,9,16,33,17,34,29,32)(4,18,22,10,21,48,23,37)(5,11,25,40,26,41,28,39)(14,42,19,30,44,35,50,58)(15,38,45,59,24,31,46,60)(20,51,53,36,54,47,55,63)(43,62,52,64,56,61,49,57);(1,3,7,16,6,17,8,29)(2,9,13,32,12,34,27,33)(4,19,22,44,21,50,23,14)(5,24,25,46,26,15,28,45)(10,35,18,30,37,42,48,58)(11,38,39,60,41,31,40,59)(20,52,53,56,54,49,55,43)(36,62,51,57,63,61,47,64);(1,4,8,23,6,21,7,22)(2,10,27,48,12,37,13,18)(3,14,16,19,17,44,29,50)(5,20,25,53,26,54,28,55)(9,30,32,42,34,58,33,35)(11,36,39,51,41,63,40,47)(15,43,46,49,24,56,45,52)(31,57,60,62,38,64,59,61);(1,5,7,28,6,26,8,25)(2,11,13,40,12,41,27,39)(3,15,29,45,17,24,16,46)(4,20,22,55,21,54,23,53)(9,31,33,59,34,38,32,60)(10,36,18,47,37,63,48,51)(14,43,50,52,44,56,19,49)(30,57,35,61,58,64,42,62)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=ef,d^2=efi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344727);