Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C42 · Schur multiplier (H2) = C213×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C2 | C24×C42 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
12828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(5,6,8,7)(9,10,15,13)(11,18,12,16)(14,21,22,17)(19,23,20,24);(5,6,8,7)(9,11,15,12)(10,16,13,18)(14,19,22,20)(17,23,21,24);(9,10,15,13)(11,16,12,18)(14,17,22,21)(19,23,20,24);(9,12,15,11)(10,16,13,18)(14,20,22,19)(17,23,21,24);(1,3,4,2)(5,7,8,6)(9,13,15,10)(11,16,12,18)(14,21,22,17)(19,23,20,24);(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=g,e^2=h,b^a=bg,f^a=fg,e^b=eg,d^c=dg260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532418);