Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,11,12,10)(13,14)(15,22)(16,20)(17,26)(18,21)(19,23)(24,27)(25,28);(2,3)(6,9)(10,11)(13,15,17,23,18,24,16,25)(14,19,20,22,21,28,26,27);(1,2,4,3)(5,7,8,12)(6,10,9,11);(1,3,4,2);(13,16,18,17)(14,20,21,26)(15,23,24,25)(19,27,28,22)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=gh,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374040);