Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93852 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7089 |
| commutator | C2×M32 | C2×C4 |
| Frattini, cosocle | #2150 | C22 |
| prime radical | C23 | #360 |
| LCS | G ▷ C2×M32 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2150 ◁ G | |
| LECS | G ▷ #2150 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2)(3,17,19,49,22,52,34,51)(4,23,26,46)(5,13)(6,33)(7,15)(8,14)(9,16)(10,18,35,54,39,56,38,62)(11,27,41,31)(12,32)(20,40,21,50,53,64,58,63)(24,45,55,36,60,37,57,48)(25,47)(28,43,61,44)(29,42,30,59);(1,3,7,19,9,22,8,34)(2,10,14,38,16,39,15,35)(4,24,29,55,26,60,30,57)(5,21,32,53,25,58,6,20)(11,40,43,63,41,64,44,50)(12,45,13,48,33,37,47,36)(17,23,51,59,52,46,49,42)(18,31,54,61,56,27,62,28)Polycyclic presentation:
2^9|a^2=d,b^2=g,c^2=fh,d^2=gh,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=defhi,e^b=eh,f^b=fhi,h^b=hi,d^c=dghi,e^c=ei38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1048);