Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,28,22,27,21,26)(19,29,30,32,20,24,31,25);(1,2)(3,4)(9,11,12,16)(10,13,14,15)(17,19,22,20)(18,24,27,29)(21,31,23,30)(25,26,32,28);(2,4)(17,20,22,19)(18,25,27,32)(21,31,23,30)(24,26,29,28);(5,7,8,6)(10,14)(13,15)(17,21,22,23)(18,26,27,28)(19,30,20,31)(24,25,29,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=i,d^2=fhi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284742);