Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #466 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1808 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(5,6,10,16,12,17,11,34)(7,18,22,46,24,47,23,56)(8,25,28,59,30,40,29,61)(9,31,32,43,33,62,19,15)(13,35,37,55,38,48,45,21)(14,39,41,26,42,60,58,27)(20,49,52,64,54,66,53,36)(44,65,67,50,57,63,68,51);(1,2)(3,4)(5,7,19,48,11,23,33,55,12,24,32,35,10,22,9,21)(6,13,31,46,16,37,43,47,17,38,62,56,34,45,15,18)(8,20,39,65,29,53,27,51,30,54,60,63,28,52,26,50)(14,36,61,67,41,49,25,57,42,64,59,68,58,66,40,44);(2,4)(5,8,9,26,10,28,32,60,12,30,33,27,11,29,19,39)(6,14,15,40,34,58,62,59,17,42,43,25,16,41,31,61)(7,20,21,50,22,52,35,63,24,54,55,51,23,53,48,65)(13,36,18,44,45,66,56,68,38,64,47,57,37,49,46,67)Polycyclic presentation:
2^9|a^2=g,b^2=dh,c^2=d,d^2=hi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56097);