Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383160 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14357 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1893 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)(3,14)(4,20,23,36)(5,24,28,11)(6,29)(7,12)(8,13)(9,17)(10,22,37,30)(15,31,46,34)(16,38,48,43)(18,33)(19,40)(21,51)(25,32,47,58)(26,39,27,41)(35,53)(42,45,50,49)(44,57)(52,60)(54,61)(55,56)(59,63)(62,64);(1,3,8,19)(2,9,13,33)(4,15,23,46)(5,25,28,47)(6,18,7,17)(10,34,37,31)(11,38,24,43)(12,40,29,14)(16,27,48,26)(20,50,36,42)(21,52,54,64)(22,49,30,45)(32,41,58,39)(35,60,56,62)(44,55,63,53)(51,59,61,57);(1,4)(2,10)(3,15)(5,21)(6,22)(7,30)(8,23)(9,31)(11,35)(12,36)(13,37)(14,42)(16,44)(17,45)(18,49)(19,46)(20,29)(24,56)(25,52)(26,53)(27,55)(28,54)(32,57)(33,34)(38,62)(39,51)(40,50)(41,61)(43,60)(47,64)(48,63)(58,59);(1,5)(2,11)(3,16)(4,21)(6,27)(7,26)(8,28)(9,32)(10,35)(12,41)(13,24)(14,43)(15,44)(17,47)(18,25)(19,48)(20,51)(22,55)(23,54)(29,39)(30,53)(31,57)(33,58)(34,59)(36,61)(37,56)(38,40)(42,60)(45,64)(46,63)(49,52)(50,62)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal quotient of | #72239049 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290357);