Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C22×D8 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,22)(15,23,26,42)(16,27,31,41)(17,32,29,21)(19,34,36,24)(20,37,39,30)(25,43,33,38)(28,40,44,35);(1,2)(3,4)(6,9)(10,12);(2,4)(13,15,18,26)(14,19,22,36)(16,28,31,44)(17,33,29,25)(20,38,39,43)(21,40,32,35)(23,41,42,27)(24,37,34,30);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,f^c=fh,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406429);