Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #272667 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11,12,9)(8,10)(13,14)(15,19)(16,18)(17,21)(20,24)(22,25)(23,27)(26,28);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(14,18)(16,24)(19,25)(20,21)(22,28)(26,27);(2,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,23)(14,19,21,27)(16,22,24,28)(18,25,20,26);(5,7)(6,9)(8,12)(10,11)(14,18)(16,24)(19,25)(20,21)(22,28)(26,27)Polycyclic presentation:
2^9|a^2=f,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,e^d=ei,f^d=fhi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 179871);