Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187159 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#538 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,11,27,34,57,60)(3,13,16,43,15,48,25,56)(4,19,22,51,21,37,53,62)(6,9,26,33,7,12,28,40)(8,29,32,14,44,46,39,45)(10,35,38,24,41,55,64,23)(17,42,49,63,18,36,50,61)(20,30,52,58,47,31,54,59);(1,3,6,17)(2,8,9,30)(4,20,23,45)(5,25,26,50)(7,18,27,15)(10,36,19,48)(11,39,33,59)(12,31,34,44)(13,41,42,37)(14,21,47,24)(16,28,49,57)(22,54,35,46)(29,53,52,55)(32,40,58,60)(38,63,51,43)(56,64,61,62);(2,9)(4,21)(5,26)(8,30)(10,37)(12,34)(13,36)(14,45)(16,49)(19,41)(20,47)(22,55)(23,24)(25,50)(28,57)(29,54)(31,44)(35,53)(38,64)(42,48)(43,56)(46,52)(51,62)(61,63);(2,9)(8,30)(10,19)(11,33)(12,34)(13,42)(31,44)(32,58)(36,48)(37,41)(38,51)(39,59)(40,60)(43,63)(56,61)(62,64)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94545);