Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,15,16)(11,13,18,24)(12,21,23,17)(14,19,20,22);(1,2,3,4)(9,11)(10,13)(12,22)(14,21)(15,18)(16,24)(17,20)(19,23);(6,8)(9,12,15,23)(10,17,16,21)(11,20,18,14)(13,19,24,22);(9,13,15,24)(10,18,16,11)(12,20,23,14)(17,19,21,22);(9,14)(10,19)(11,12)(13,21)(15,20)(16,22)(17,24)(18,23)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481370);