Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×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 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(5,6,7,8)(17,18)(19,26)(20,25)(21,31)(22,28)(23,27)(24,29)(30,32);(2,4)(9,10)(11,14)(12,13)(15,16)(17,19,24,30)(18,25,29,31)(20,22,21,23)(26,27,32,28);(10,13)(14,16)(17,20,24,21)(18,26,29,32)(19,23,30,22)(25,28,31,27);(2,4)(9,11,12,15)(10,14,13,16)(17,21,24,20)(18,26,29,32)(19,22,30,23)(25,28,31,27)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi,e^d=ei,f^d=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270487);