Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437793 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,20,16,11,15,19)(12,17,22,23,13,18,24,21);(2,5)(6,8)(9,11)(10,16)(12,21)(13,23)(14,20)(15,19)(17,24)(18,22);(2,5)(6,8)(9,12)(10,17)(11,18)(13,16)(14,22)(15,24)(19,21)(20,23);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,17)(12,16)(14,24)(15,22)(19,23)(20,21);(9,14,16,15)(10,19,11,20)(12,22,13,24)(17,21,18,23)Polycyclic presentation:
2^9|a^2=fg,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6357138);