Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427949 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15376 |
| 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 | |
1235 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)(3,8)(4,17,19,38)(5,21,23,10)(6,11)(7,12)(9,20,34,26)(13,41,43,48)(14,30,22,35)(15,31)(16,32)(18,49)(24,39,27,36)(25,37)(28,40)(29,44,56,46)(33,50)(42,62)(45,57,47,58)(51,59)(52,53)(54,61)(55,64)(60,63);(1,3,7,16)(2,8,12,32)(4,13,19,43)(5,22,23,14)(6,25,28,15)(9,29,34,56)(10,35,21,30)(11,37,40,31)(17,48,38,41)(18,42,51,54)(20,46,26,44)(24,45,27,47)(33,55,53,60)(36,57,39,58)(49,62,59,61)(50,64,52,63);(1,4)(2,9)(3,13)(5,18)(6,26)(7,19)(8,29)(10,33)(11,38)(12,34)(14,42)(15,46)(16,43)(17,40)(20,28)(21,53)(22,54)(23,51)(24,52)(25,44)(27,50)(30,55)(31,48)(32,56)(35,60)(36,49)(37,41)(39,59)(45,64)(47,63)(57,61)(58,62);(1,5,6,27,7,23,28,24)(2,10,11,39,12,21,40,36)(3,14,15,47,16,22,25,45)(4,18,26,50,19,51,20,52)(8,30,31,58,32,35,37,57)(9,33,38,59,34,53,17,49)(13,42,46,63,43,54,44,64)(29,55,48,62,56,60,41,61)Polycyclic presentation:
2^9|a^2=e,b^2=i,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335146);