Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 | #36213 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4633 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)(11,14)(12,16)(13,17)(15,18)(19,24)(20,23)(21,22);(2,4)(9,11,14,10)(12,20,13,22)(15,18,24,19)(16,21,17,23);(6,8)(9,12,14,13)(10,16,11,17)(15,23,24,21)(18,20,19,22);(10,11)(12,21)(13,23)(15,24)(16,20)(17,22);(9,13,14,12)(10,17,11,16)(15,21,24,23)(18,22,19,20)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,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.
| maximal subgroup of | #72313514, #72313597, #72313685 |
| maximal normal subgroup of | #72313514, #72313597, #72313685 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480703);