Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395154 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(6,7)(17,18)(19,23)(20,28)(21,22)(24,26)(25,27)(29,31)(30,32);(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(18,22)(23,25)(24,31)(28,32);(2,4)(5,7,8,6)(10,13)(14,16)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,28,31,32);(9,11,12,15)(10,14,13,16)(17,20,27,29,21,30,19,26)(18,24,23,32,22,31,25,28)Polycyclic presentation:
2^9|b^2=ei,c^2=e,d^2=ghi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321318);