Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #388 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13842 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
957 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)(3,15,20,32)(4,21)(5,11)(6,12)(7,13)(8,14)(9,18,33,19)(10,25)(16,42,45,30)(17,41,48,56)(22,50,44,34)(23,49)(24,36)(26,40)(27,37)(28,38)(29,39)(31,46,57,47)(35,53)(43,61,63,55)(51,64,62,58)(52,59)(54,60);(1,3,6,18,8,20,7,19)(2,9,13,32,14,33,12,15)(4,22,25,16,26,44,24,45)(5,17,27,46,29,48,28,47)(10,34,21,42,36,50,40,30)(11,31,38,56,39,57,37,41)(23,51,53,43,54,62,52,63)(35,58,49,61,59,64,60,55);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,30)(11,35)(12,21)(13,40)(14,36)(15,34)(17,43)(18,22)(19,44)(20,45)(27,52)(28,53)(29,54)(31,55)(32,50)(33,42)(37,49)(38,60)(39,59)(41,58)(46,51)(47,62)(48,63)(56,64)(57,61);(1,5,6,27,8,29,7,28)(2,11,12,37,14,39,13,38)(3,17,18,46,20,48,19,47)(4,23,24,52,26,54,25,53)(9,31,15,41,33,57,32,56)(10,35,21,49,36,59,40,60)(16,43,22,51,45,63,44,62)(30,55,34,58,42,61,50,64)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,h^b=hi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265618);