Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #197026 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60961 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6,8,11)(7,13,14,16)(17,18)(19,22,26,31)(20,25)(21,24)(23,27,32,28)(29,30);(1,2,4,11,5,8,12,6)(3,7,10,16,9,14,15,13)(17,19)(18,22,24,31)(20,28)(21,26)(23,30,32,25)(27,29);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,19)(18,23)(20,28)(21,26)(22,30)(24,32)(25,31)(27,29);(3,9)(7,14)(10,15)(13,16)(19,26)(22,31)(23,32)(27,28)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,f^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 104222);