Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(10,12)(13,14,19,24,18,23,17,22)(15,21,25,28,16,20,26,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,18)(17,26)(19,25)(21,23)(22,28)(24,27);(1,2,3,4)(13,16,18,15)(14,21,23,20)(17,25,19,26)(22,27,24,28);(6,9)(10,12)(15,16)(20,21)(25,26)(27,28);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,22,23,24)(15,25,16,26)(20,27,21,28)Polycyclic presentation:
2^9|a^2=h,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ci,d^b=df,e^b=ei,d^c=di,e^c=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287561);