Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| 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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,18,23,19,22,26,21,25)(20,24,31,27,30,32,29,28);(1,2)(3,4)(9,11,13,12)(10,14,16,15)(17,19,22,25)(18,21,26,23)(20,28,30,27)(24,31,32,29);(2,4)(9,11,13,12)(10,15,16,14)(17,20,22,30)(18,24,26,32)(19,27,25,28)(21,29,23,31);(9,12,13,11)(10,14,16,15)(18,25)(19,26)(21,23)(24,28)(27,32)(29,31)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=fi,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292813);