Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #427 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1808 |
| prime radical | C24 | C2×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(2,4)(5,6,11,18,13,19,12,36)(7,20,24,48,25,49,35,38)(8,26,29,59,31,60,30,43)(9,17,10,34,33,46,32,16)(14,23,39,22,41,55,40,56)(15,42,44,61,45,27,58,28)(21,50,53,64,54,37,62,63)(47,51,66,68,67,65,57,52);(5,7,10,23,11,35,33,56,13,25,32,55,12,24,9,22)(6,14,17,38,36,39,16,20,19,41,46,48,18,40,34,49)(8,21,28,52,29,62,42,65,31,54,61,68,30,53,27,51)(15,37,26,57,58,63,43,47,45,50,60,66,44,64,59,67);(1,2,3,4)(5,8,9,27,12,30,32,61,13,31,33,42,11,29,10,28)(6,15,34,59,18,44,46,60,19,45,16,43,36,58,17,26)(7,21,22,51,24,53,55,68,25,54,56,65,35,62,23,52)(14,37,49,67,40,64,48,66,41,50,20,47,39,63,38,57)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=eh,d^2=h,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,f^b=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54293);