Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437478 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,16)(14,23)(15,17)(18,19)(20,22)(21,24);(9,10)(11,18)(12,21)(13,16)(14,23)(15,17)(19,22)(20,24);(2,5)(6,8)(9,11)(10,12)(13,21)(14,19)(15,20)(16,24)(17,22)(18,23);(1,3,4,7)(2,6,5,8)(9,12)(10,11)(13,18)(14,24)(15,22)(16,19)(17,20)(21,23);(9,13,15,14)(10,16,17,23)(11,19,20,21)(12,18,22,24)Polycyclic presentation:
2^9|d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355866);