Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432833 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18,20,24,22,25,21,26)(19,27,28,32,30,31,29,23);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,23,25,32)(20,29,21,28)(24,31,26,27);(11,15)(13,16)(18,24)(20,21)(23,31)(25,26)(27,32)(28,29);(1,3,4,7)(2,6,5,8)(10,14)(13,16);(18,25)(23,32)(24,26)(27,31)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341407);