Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7606 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6)(7,11,12,9)(8,10)(15,18,19,20)(21,22,23,25,24,26,27,28);(1,2,3,4)(13,14,17,16)(15,18,19,20);(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,20)(17,19)Polycyclic presentation:
2^9|a^2=f,b^2=dh,c^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,e^a=eg,e^c=eh,f^c=fgh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5984);