Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629016 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35251 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4607 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)(3,4)(5,6)(7,8)(9,10,14,17)(11,19,21,22)(12,16,15,24)(13,20,23,18);(9,11,13,16,14,21,23,24)(10,15,18,19,17,12,20,22);(2,4)(9,12)(10,16)(11,20)(13,22)(14,15)(17,24)(18,21)(19,23);(6,8)(10,17)(11,21)(16,24)(18,20);(11,21)(12,15)(16,24)(19,22)Polycyclic presentation:
2^9|a^2=i,b^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,d^b=di,e^b=ei,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72315401, #72315509 |
| maximal normal subgroup of | #72315401, #72315509 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536219);