Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#385 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,11,13,16)(4,9,5,8)(7,14,15,12)(17,18,21,25,23,20,22,26)(19,27,30,24,31,32,29,28);(1,3)(2,7)(4,14)(5,12)(6,13)(8,16)(9,11)(10,15)(17,19,22,29,23,31,21,30)(18,24,26,27,20,28,25,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,16)(10,15)(17,20)(18,23)(19,28)(21,26)(22,25)(24,31)(27,30)(29,32);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,11,15,16)(17,21,23,22)(18,25,20,26)(19,29,31,30)(24,27,28,32)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fh,d^b=dh,e^b=eh,f^b=fi,e^c=ei,f^c=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 99715);