Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363904 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2297 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13958 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1103 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)(3,13,17,36)(4,18,21,9)(5,22,24,51)(6,11)(7,12)(8,16,28,43)(10,23,32,52)(14,34)(15,40)(19,46)(20,45,49,31)(25,33)(26,53)(27,42)(29,38)(30,47)(35,58)(37,55,61,54)(39,56)(41,59)(44,62)(48,50)(57,60,63,64);(1,3)(2,8,12,28)(4,14)(5,15)(6,16)(7,17)(9,29,18,26)(10,27,32,40)(11,13,33,36)(19,37)(20,38)(21,39)(22,35,51,59)(23,41)(24,42)(25,43)(30,55,44,54)(31,56,45,34)(46,57,50,63)(47,60)(48,61)(49,53)(52,58)(62,64);(1,4,6,20,7,21,25,49)(2,9,11,31,12,18,33,45)(3,14,16,38,17,39,43,53)(5,19,23,47,24,48,52,62)(8,26,13,34,28,29,36,56)(10,30,22,50,32,44,51,46)(15,37,41,60,42,61,58,64)(27,54,35,57,40,55,59,63);(1,5)(2,10,12,32)(3,15)(4,19)(6,23)(7,24)(8,27,28,40)(9,30,18,44)(11,22,33,51)(13,35,36,59)(14,37)(16,41)(17,42)(20,47)(21,48)(25,52)(26,54,29,55)(31,50,45,46)(34,57,56,63)(38,60)(39,61)(43,58)(49,62)(53,64)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=e,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,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, 271255);