Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806551 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13);(2,5)(3,7)(17,18,23,27)(19,28,30,26)(20,25,22,31)(21,24,29,32);(9,11,12,16)(10,13,14,15)(17,19,22,21,23,30,20,29)(18,24,25,28,27,32,31,26);(2,5)(6,8)(17,20,23,22)(18,25,27,31)(19,29,30,21)(24,28,32,26);(1,3,4,7)(2,6,5,8)(17,21,22,30,23,29,20,19)(18,26,25,24,27,28,31,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=i,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, 6725428);