Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454968 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1269 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,8)(9,10)(11,19,20,15)(12,16,21,24)(13,22)(14,18)(17,23);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,13)(14,20)(16,17)(18,19)(21,23)(22,24);(3,7)(6,8)(9,12)(10,16,18,24)(11,13)(14,21)(15,22,19,17)(20,23);(1,3,4,7)(2,6,5,8)(9,13)(10,17,18,22)(11,12)(14,23)(15,16,19,24)(20,21)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=de,f^c=fi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376078);