Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #450130 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15958 |
| 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 | |
994 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)(3,13,17,8)(4,18)(5,10,24,35)(6,25)(7,11)(9,21)(12,27)(14,39,19,47)(15,29)(16,30,45,38)(20,46,50,59)(22,37)(23,51,53,36)(26,33)(28,41,31,44)(32,52,57,49)(34,43)(40,61)(42,58)(48,60)(54,62)(55,64)(56,63);(1,3,6,16,7,17,27,45)(2,8,25,38,11,13,12,30)(4,19,21,44,22,14,26,41)(5,15,23,42,24,43,53,64)(9,31,37,47,33,28,18,39)(10,34,51,55,35,29,36,58)(20,48,49,63,50,40,52,62)(32,54,59,60,57,56,46,61);(1,4,7,22)(2,9,11,33)(3,14,17,19)(5,20,24,50)(6,26,27,21)(8,28,13,31)(10,32,35,57)(12,37,25,18)(15,40,43,48)(16,44,45,41)(23,52,53,49)(29,54,34,56)(30,39,38,47)(36,59,51,46)(42,63,64,62)(55,60,58,61);(1,5)(2,10,11,35)(3,15)(4,20)(6,23)(7,24)(8,29,13,34)(9,32,33,57)(12,36,25,51)(14,40)(16,42)(17,43)(18,46,37,59)(19,48)(21,49)(22,50)(26,52)(27,53)(28,54,31,56)(30,55,38,58)(39,60,47,61)(41,62)(44,63)(45,64)Polycyclic presentation:
2^9|a^2=eh,b^2=ef,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357327);