Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518904 |
| 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 | #30906 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,7)(8,11,12,10)(13,14)(15,24)(16,21)(17,27)(18,28)(19,23)(20,26)(22,25);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,15,18,26,19,17,16,25)(14,20,21,24,23,22,28,27);(5,8,9,12)(6,10,7,11)(13,16,19,18)(14,21,23,28)(15,25,17,26)(20,24,22,27);(2,4)(6,7)(10,11);(13,17,18,25,19,15,16,26)(14,22,21,27,23,20,28,24)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=gi,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7452959);