Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401983 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22⋊#401 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
873 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)(3,13,16,8)(4,9,20,33)(5,22)(6,25)(7,11)(10,23)(12,27)(14,31)(15,41,42,54)(17,30,44,38)(18,28)(19,48,49,60)(21,52,51,34)(24,36)(26,35)(29,43,37,45)(32,53,59,50)(39,61)(40,56)(46,64)(47,58)(55,63)(57,62);(1,3)(2,8)(4,18,20,14)(5,15)(6,17)(7,16)(9,31,33,28)(10,29)(11,13)(12,30)(19,46,49,39)(21,47,51,40)(22,54)(23,43)(24,42)(25,38)(26,45)(27,44)(32,57,59,55)(34,58,52,56)(35,37)(36,41)(48,61,60,64)(50,63,53,62);(1,4)(2,9,11,33)(3,14)(5,19)(6,21)(7,20)(8,28,13,31)(10,32,35,59)(12,34,25,52)(15,39)(16,18)(17,40)(22,48,36,60)(23,50)(24,49)(26,53)(27,51)(29,55,37,57)(30,56,38,58)(41,61,54,64)(42,46)(43,62)(44,47)(45,63);(1,5,7,24)(2,10,11,35)(3,15,16,42)(4,19,20,49)(6,26,27,23)(8,29,13,37)(9,32,33,59)(12,36,25,22)(14,39,18,46)(17,45,44,43)(21,53,51,50)(28,55,31,57)(30,41,38,54)(34,60,52,48)(40,63,47,62)(56,61,58,64)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309180);