Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417575 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,10)(11,12)(13,14,17,20,16,19,24,25)(15,21,23,28,22,18,26,27);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,22)(17,23)(19,21)(20,27)(24,26)(25,28);(2,4)(5,7,9,8)(6,11,10,12)(13,14)(15,21)(16,19)(17,25)(18,22)(20,24)(23,27)(26,28);(5,8,9,7)(6,11,10,12)(15,22)(17,24)(18,21)(20,25)Polycyclic presentation:
2^9|a^2=f,c^2=ei,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327561);