Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #376205 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14175 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1498 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)(3,13)(4,18,22,9)(5,23)(6,27)(7,12)(8,16)(10,25)(11,24)(14,39,19,36)(15,42)(17,38)(20,50,52,57)(21,33,28,35)(26,53)(29,46,47,41)(30,45)(31,43)(32,51,48,55)(34,54)(37,44)(40,61,49,56)(58,62,60,64)(59,63);(1,3,6,16,7,17,24,43)(2,8,11,13,12,31,27,38)(4,19,21,46,22,14,28,41)(5,15,25,44,26,45,54,63)(9,29,33,39,18,47,35,36)(10,30,23,37,34,42,53,59)(20,49,51,64,52,40,55,62)(32,56,57,60,48,61,50,58);(1,4)(2,9)(3,14)(5,20)(6,28)(7,22)(8,29)(10,32)(11,35)(12,18)(13,36)(15,40)(16,46)(17,19)(21,24)(23,50)(25,55)(26,52)(27,33)(30,56)(31,47)(34,48)(37,58)(38,39)(41,43)(42,61)(44,64)(45,49)(51,54)(53,57)(59,60)(62,63);(1,5,24,54,7,26,6,25)(2,10,27,53,12,34,11,23)(3,15,43,63,17,45,16,44)(4,20,21,51,22,52,28,55)(8,30,38,59,31,42,13,37)(9,32,33,57,18,48,35,50)(14,40,41,62,19,49,46,64)(29,56,39,60,47,61,36,58)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=ehi,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,h^c=hi,f^d=fi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283402);