Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(7,11)(13,14,17,20,16,18,24,25)(15,21,23,28,22,19,26,27);(5,6)(7,10)(8,9)(11,12)(14,18)(19,21)(20,25)(27,28);(1,2,3,4)(6,9)(10,12);(5,7,8,11)(6,10,9,12)(13,15,16,22)(14,19,18,21)(17,23,24,26)(20,27,25,28);(15,22)(19,21)(23,26)(27,28)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6282204);