Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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 | |
1230 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)(12,16)(13,14)(17,18)(19,25)(20,29)(21,24)(22,31)(23,27)(26,32)(28,30);(1,2)(3,4)(5,7,8,6)(10,14)(15,16)(17,19)(18,23)(20,30)(21,27)(22,28)(24,25)(26,31)(29,32);(2,4)(9,11,13,12)(10,15,14,16)(17,20,29,18,21,31,22,24)(19,26,32,25,27,30,28,23);(9,12,13,11)(10,15,14,16)(18,24)(19,27)(20,31)(28,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=ef,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272870);