Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343678 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62823 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,8,11,28)(4,18)(5,21)(9,26,32,52)(10,34,36,54)(12,29,13,22)(15,38)(16,44)(19,46)(20,49)(23,25)(27,35,55,37)(30,45,33,53)(31,57,58,64)(39,50)(41,43)(42,61)(47,48)(51,59,63,56)(60,62);(1,2,6,12)(3,9,17,33)(4,10,19,35)(5,22,21,28)(7,13,24,11)(8,25,29,23)(14,30,40,32)(15,31,43,59)(16,45,44,52)(18,37,46,36)(20,34,48,55)(26,50,53,39)(27,49,54,47)(38,56,41,58)(42,57,62,63)(51,61,64,60);(3,14)(4,19)(9,30)(10,35)(15,41)(16,39)(17,40)(18,46)(20,48)(26,45)(27,54)(31,58)(32,33)(34,55)(36,37)(38,43)(42,61)(44,50)(47,49)(51,63)(52,53)(56,59)(57,64)(60,62);(1,3,6,17)(2,9,12,33)(4,15,19,43)(5,16,21,44)(7,14,24,40)(8,26,29,53)(10,31,35,59)(11,32,13,30)(18,38,46,41)(20,42,48,62)(22,45,28,52)(23,39,25,50)(27,51,54,64)(34,57,55,63)(36,58,37,56)(47,60,49,61);(1,4,6,19)(2,10,12,35)(3,15,17,43)(5,20,21,48)(7,18,24,46)(8,27,29,54)(9,31,33,59)(11,36,13,37)(14,38,40,41)(16,42,44,62)(22,34,28,55)(23,47,25,49)(26,51,53,64)(30,56,32,58)(39,60,50,61)(45,57,52,63)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,e^2=h,f^2=i,b^a=bf,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250880);