Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | #33516 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,18,15,20,31,19)(11,16,22,33,23,34,13,29)(12,24,26,32,28,40,27,36)(17,35,37,25,38,30,39,21);(1,3)(2,5)(4,8)(6,7)(9,11,14,22,15,23,31,13)(10,16,18,33,20,34,19,29)(12,25,26,30,28,21,27,35)(17,36,37,24,38,32,39,40);(3,8)(5,7)(9,12)(10,17)(11,21)(13,30)(14,26)(15,28)(16,32)(18,37)(19,39)(20,38)(22,35)(23,25)(24,29)(27,31)(33,40)(34,36);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=g,b^2=g,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cei,f^c=fh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404018);