Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1566 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)(5,6,7,8)(17,18,21,25,20,24,22,28)(19,23,27,31,26,30,29,32);(6,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,28,20,24,21,25)(19,23,29,32,26,30,27,31);(2,4)(10,13)(14,16)(17,19)(18,23)(20,26)(21,27)(22,29)(24,30)(25,31)(28,32);(6,8)(9,11,12,15)(10,14,13,16)(19,26)(23,30)(27,29)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278123);