Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187177 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62830 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4,10,12)(5,13)(6,8)(7,15)(9,16,14,11)(17,18,21,25,22,26,20,30)(19,27,28,23,29,24,31,32);(1,3)(2,4)(5,14)(6,10)(7,16)(8,12)(9,15)(11,13)(17,19)(18,23)(20,31)(21,28)(22,29)(24,25)(26,32)(27,30);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,14)(15,16)(17,19)(18,24)(20,28)(21,31)(22,29)(23,25)(26,27)(30,32);(1,5,6,15)(2,7,8,13)(3,9,10,14)(4,11,12,16)(17,20,22,21)(18,25,26,30)(19,28,29,31)(23,27,32,24)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94509);