Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #94978 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#95 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6,9,12,10,7,8,11)(13,14)(15,18,21,25)(16,19)(17,24)(20,23)(22,28,27,26);(5,7,10,6)(8,12,9,11)(13,15,17,22,16,21,23,27)(14,18,20,26,19,25,24,28);(1,2,3,4)(5,8,10,9)(6,11,7,12)(14,19)(18,25)(20,24)(26,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=eh,d^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=dh,f^b=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 16349);