Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363904 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2297 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13960 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1351 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,7,13,6,12)(3,15,18,33,19,37,28,9)(4,20,23,38,24,10,29,35)(5,25,26,36,27,39,30,11)(16,40,44,31,21,51,43,49)(17,45,48,41,47,32,46,59)(22,53,56,50,55,57,54,34)(42,61,62,64,52,58,63,60);(1,3,8,28,7,19,6,18)(2,9,12,15,13,33,14,37)(4,21,29,43,24,16,23,44)(5,17,30,48,27,47,26,46)(10,31,35,40,20,49,38,51)(11,32,25,41,36,45,39,59)(22,52,56,62,55,42,54,63)(34,58,57,60,50,61,53,64);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,29)(9,31)(11,34)(12,35)(13,20)(14,38)(15,40)(17,42)(18,43)(19,21)(25,57)(26,54)(27,55)(28,44)(30,56)(32,58)(33,49)(36,50)(37,51)(39,53)(41,60)(45,61)(46,62)(47,52)(48,63)(59,64);(1,5,8,30,7,27,6,26)(2,11,12,25,13,36,14,39)(3,17,28,48,19,47,18,46)(4,22,29,56,24,55,23,54)(9,32,15,41,33,45,37,59)(10,34,35,57,20,50,38,53)(16,42,44,63,21,52,43,62)(31,58,40,60,49,61,51,64)Polycyclic presentation:
2^9|a^2=h,b^2=eg,d^2=eg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271347);