Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806551 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2013 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)(5,6,9,11)(7,12,8,10)(13,14,17,21)(15,23,24,19)(16,20,25,27)(18,22,26,28);(5,6)(7,10)(8,12)(9,11)(13,15,17,24)(14,19,21,23)(16,26,25,18)(20,22,27,28);(2,4)(13,16,18,15,17,25,26,24)(14,20,22,23,21,27,28,19);(5,7,9,8)(6,10,11,12)(14,21)(19,23)(20,27)(22,28);(5,8,9,7)(6,10,11,12)(13,16,26,24,17,25,18,15)(14,20,28,19,21,27,22,23)Polycyclic presentation:
2^9|a^2=f,b^2=i,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,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718224);