Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #496285 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #21758 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #60461 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12,14,7)(4,8)(5,20)(6,9)(10,17)(11,18)(13,24,16,21)(15,23,30,28)(19,25)(22,29,27,31)(26,32);(1,3)(2,7)(4,16)(5,15)(6,14)(8,24)(9,12)(10,27)(11,23)(13,19)(17,31)(18,30)(20,28)(21,25)(22,26)(29,32);(2,8)(4,17)(5,18)(7,21)(9,25)(10,11)(12,24)(13,29)(15,30)(16,31)(19,32)(20,26)(22,23)(27,28);(1,3)(2,7)(4,13)(5,15)(6,14)(8,21)(9,12)(10,22)(11,23)(16,19)(17,29)(18,30)(20,28)(24,25)(26,27)(31,32)Polycyclic presentation:
2^9|a^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=ce,f^b=fi,g^b=gi,f^c=fh,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403481);