Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C23×#41 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,9,10,8,12,7,11)(13,14)(15,23)(16,20)(17,21)(18,22)(19,25)(24,28)(26,27);(5,7,8,9)(6,10,12,11)(13,15,18,25)(14,19,22,23)(16,24,17,26)(20,27,21,28);(2,4)(5,7,8,9)(6,11,12,10)(13,16,18,17)(14,20,22,21)(15,24,25,26)(19,27,23,28);(5,7,8,9)(6,11,12,10)(13,17,18,16)(14,21,22,20)(15,24,25,26)(19,27,23,28);(6,12)(10,11)(14,22)(19,23)(20,21)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293477);