Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| prime radical | #62815 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,20)(14,17)(18,23)(19,24)(21,22);(11,18,19,21)(15,22,24,23);(2,5)(6,8)(9,11)(10,15)(12,18)(13,21)(14,19)(16,23)(17,24)(20,22);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,18)(13,21)(14,19)(16,23)(17,24)(20,22);(9,12,14,13)(10,16,17,20)(11,18,19,21)(15,23,24,22)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,f^c=fi,h^c=hi,f^d=fi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6357211);