Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2250 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(13,14,21,24)(15,25,23,18)(16,20,27,26)(17,28,22,19);(1,2)(3,4)(6,9)(10,11)(13,15,19,27,20,17,18,14,21,23,28,16,26,22,25,24);(2,4)(13,16,18,15,26,14,19,22,21,27,25,23,20,24,28,17);(5,7,8,12)(6,10,9,11)(13,17,19,14,20,23,18,16,21,22,28,24,26,15,25,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fh,d^2=ehi,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg,d^b=di,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411201);