Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #361285 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2290 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13894 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1343 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)(3,15,19,39)(4,20,24,10)(5,25,28,38)(6,12)(7,14)(8,13)(9,18,33,29)(11,26,37,27)(16,34)(17,46)(21,50)(22,52)(23,40,30,36)(31,44)(32,48)(35,53)(41,45)(42,56)(43,58,63,64)(47,60)(49,57)(51,59,62,61)(54,55);(1,3)(2,9,13,33)(4,21)(5,17)(6,18)(7,29)(8,19)(10,34,20,41)(11,32,37,46)(12,15,14,39)(16,23)(22,51)(24,44)(25,42,38,60)(26,47)(27,56)(28,48)(30,45)(31,40,50,36)(35,59,49,61)(43,53)(52,64,55,58)(54,62)(57,63);(1,4)(2,10)(3,16)(5,22)(6,23)(7,30)(8,24)(9,31)(11,35)(12,36)(13,20)(14,40)(15,41)(17,43)(18,44)(19,45)(21,29)(25,55)(26,53)(27,57)(28,54)(32,58)(33,50)(34,39)(37,49)(38,52)(42,61)(46,64)(47,62)(48,63)(51,56)(59,60);(1,5)(2,11,13,37)(3,17)(4,22)(6,26)(7,27)(8,28)(9,32,33,46)(10,35,20,49)(12,25,14,38)(15,42,39,60)(16,43)(18,47)(19,48)(21,51)(23,53)(24,54)(29,56)(30,57)(31,58,50,64)(34,59,41,61)(36,55,40,52)(44,62)(45,63)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268481);