Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #42445 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(6,10)(7,8);(5,6)(7,12)(8,11)(9,10)(13,14)(15,24)(16,21)(17,25)(18,28)(19,23)(20,22)(26,27);(2,4)(5,7,9,8)(6,11,10,12)(13,15,18,22,19,25,16,26)(14,20,21,24,23,27,28,17);(5,8,9,7)(6,11,10,12)(13,16,19,18)(14,21,23,28)(15,22,25,26)(17,27,24,20);(13,17,18,20,19,24,16,27)(14,22,21,25,23,26,28,15)Polycyclic presentation:
2^9|c^2=h,d^2=fi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717809);