Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C32 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C22⋊C4 |
| commutator | C22 | C4×C32 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1374 |
| prime radical | C23 | C4×C16 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,19,23,20,24,27,33,21,25,28,34,30,36,39,43,22,26,29,35,31,37,40,44,32,38,41,45,42,46,47,48);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=g,e^2=f,g^2=h,h^2=i,b^a=bc,c^a=cf,d^b=df176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1312);