Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1862 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(5,6,11,16,12,17,34,49,13,19,9,18,10,36,35,50)(7,20,24,41,23,48,37,42,25,52,22,40,38,51,33,14)(8,26,30,58,31,47,54,46,32,60,28,59,29,15,43,45)(21,53,27,39,56,44,62,65,57,67,55,66,61,68,63,64);(1,2)(3,4)(5,7,10,23,13,25,12,38)(6,14,17,40,19,42,36,41)(8,27,29,62,32,55,31,63)(9,33,34,24,11,37,35,22)(15,44,26,53,47,68,60,67)(16,48,49,20,18,51,50,52)(21,54,56,30,57,43,61,28)(39,59,64,45,66,58,65,46);(2,4)(5,8,9,28,12,31,35,43,13,32,11,30,10,29,34,54)(6,15,16,45,17,26,49,58,19,47,18,46,36,60,50,59)(7,21,22,55,23,56,33,63,25,57,24,27,38,61,37,62)(14,39,20,44,41,65,48,67,42,66,52,68,40,64,51,53)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=dh,f^b=fhi,g^b=gh,h^b=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56259);