Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C8 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,10,16,12,19,30,18)(7,20,24,47,22,46,53,36)(8,14,28,40,26,38,56,42)(9,17,11,15,31,43,32,44)(13,25,35,52,34,54,37,23)(21,45,49,64,51,65,66,59)(27,41,29,39,55,63,57,62)(33,50,58,67,61,68,60,48);(2,4)(5,7,11,25,10,24,31,52,12,22,32,54,30,53,9,23)(6,13,17,36,18,37,44,46,19,34,43,47,16,35,15,20)(8,21,29,50,28,49,55,67,26,51,57,68,56,66,27,48)(14,33,41,59,42,60,62,65,38,61,63,64,40,58,39,45);(5,8,10,28,12,26,30,56)(6,14,16,40,19,38,18,42)(7,21,24,49,22,51,53,66)(9,27,11,29,31,55,32,57)(13,33,35,58,34,61,37,60)(15,39,43,63,44,62,17,41)(20,45,47,64,46,65,36,59)(23,48,25,50,52,67,54,68);(7,22)(8,26)(13,34)(14,38)(20,46)(23,52)(24,53)(25,54)(27,55)(28,56)(29,57)(35,37)(36,47)(39,62)(40,42)(41,63)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=be,e^a=eh,g^a=ghi,h^a=hi,d^b=di,d^c=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400259);