Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C22×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator | C22 | C2×C43 |
| socle, prime radical | C24 | C23×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6,8,7)(13,14)(15,18)(16,20)(17,19)(21,24)(22,25)(23,26)(27,28);(2,4)(13,15,16,22,17,21,23,27)(14,18,20,25,19,24,26,28);(9,10,11,12)(15,21)(18,24)(22,27)(25,28);(5,7,8,6)(14,19)(18,24)(20,26)(25,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=ef,g^2=i,b^a=be,d^a=di,c^b=ci176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260413);