Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509065 |
| 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 | C22×D8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
570 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)(7,9)(8,10)(11,12)(13,14,19,22,26,23,27,25,18,15,16,20,17,21,28,24);(1,2)(3,4)(5,7,9,6)(8,11,12,10);(2,4)(5,8,9,12)(6,10,7,11)(13,15,18,14)(16,24,19,25)(17,21,26,23)(20,27,22,28);(6,7)(10,11)(13,14,18,15)(16,25,19,24)(17,23,26,21)(20,28,22,27)Polycyclic presentation:
2^9|a^2=fi,b^2=e,c^2=ei,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417148);