Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511495 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32613 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#509 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2960 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,13,7,10,21,11)(3,14,16,34,18,33,17,8)(4,9,25,37,23,36,22,44)(6,27,26,39,30,12,40,43)(15,47,49,62,50,45,48,31)(19,41,51,32,52,46,28,61)(20,54,53,35,56,63,55,64)(24,58,59,42,57,38,29,60);(1,3)(2,8)(4,20)(5,16)(6,28)(7,18)(9,35)(10,34)(11,33)(12,41)(13,14)(15,24)(17,21)(19,26)(22,55)(23,56)(25,53)(27,46)(29,48)(30,51)(31,38)(32,43)(36,64)(37,63)(39,61)(40,52)(42,45)(44,54)(47,60)(49,59)(50,57)(58,62);(1,4,21,25,7,23,5,22)(2,9,13,44,10,36,11,37)(3,15,16,48,18,50,17,49)(6,29,40,24,30,59,26,57)(8,31,33,47,34,62,14,45)(12,42,43,58,27,60,39,38)(19,53,51,20,52,55,28,56)(32,63,41,64,61,54,46,35);(2,10)(3,16,18,17)(8,14,34,33)(9,36)(11,13)(12,27)(15,48,50,49)(19,51,52,28)(20,55,56,53)(31,45,62,47)(32,46,61,41)(35,54,64,63)(37,44)(38,58)(39,43)(42,60)Polycyclic presentation:
2^9|a^2=ef,c^2=ei,d^2=efhi,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,h^c=hi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418694);