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×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1936 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)(13,14)(15,24)(16,22)(17,21)(18,23)(19,25)(20,26)(27,28);(5,6)(7,12)(8,11)(9,10)(13,15,17,25)(14,19,21,24)(16,27,26,18)(20,28,22,23);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,15,17,26,27,25)(14,20,23,19,21,22,28,24);(5,8,9,7)(6,11,10,12)(14,21)(19,24)(20,22)(23,28);(13,16,27,25,17,26,18,15)(14,22,28,19,21,20,23,24)Polycyclic presentation:
2^9|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, 6717856);