Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392203 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2280 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14685 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1281 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)(3,15,19,31)(4,10,23,36)(5,25,27,38)(6,12)(7,14)(8,13)(9,18,32,20)(11,26,37,28)(16,39)(17,45)(21,49)(22,34,24,35)(29,44)(30,47)(33,52)(40,48)(41,60,62,53)(42,55)(43,54)(46,56)(50,57)(51,58)(59,61,64,63);(1,3,8,20,7,19,6,18)(2,9,12,15,14,32,13,31)(4,16,24,44,23,43,22,42)(5,17,28,48,27,47,26,46)(10,29,34,39,36,55,35,54)(11,30,25,40,37,45,38,56)(21,41,52,63,51,62,50,61)(33,53,49,59,57,60,58,64);(1,4)(2,10)(3,16)(5,21)(6,22)(7,23)(8,24)(9,29)(11,33)(12,34)(13,35)(14,36)(15,39)(17,41)(18,42)(19,43)(20,44)(25,49)(26,50)(27,51)(28,52)(30,53)(31,54)(32,55)(37,57)(38,58)(40,59)(45,60)(46,61)(47,62)(48,63)(56,64);(1,5,6,26,7,27,8,28)(2,11,12,25,14,37,13,38)(3,17,18,46,19,47,20,48)(4,21,22,50,23,51,24,52)(9,30,15,40,32,45,31,56)(10,33,34,49,36,57,35,58)(16,41,42,61,43,62,44,63)(29,53,39,59,55,60,54,64)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,h^b=hi,h^c=hi,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299601);