Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C24 |
| commutator | C22 | C23×C16 |
| socle | C23 | C23×C8 |
| prime radical | C22×#41 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,9)(7,11)(13,14,15,18,16,19,21,24,17,20,22,25,23,26,27,28);(5,6)(7,10)(8,9)(11,12);(2,4)(6,9)(10,12);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=g,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=ce176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403647);