Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C32 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | D4 |
| commutator | C4 | C4×C32 |
| Frattini, cosocle | C22×C32 | C22 |
| socle | C22 | C4⋊C32 |
| prime radical | C23 | M64 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 68 points:
(1,2)(3,4)(5,6,8,15,9,17,22,38,10,18,23,39,26,42,46,61,11,14,24,34,27,35,47,56,29,36,49,57,51,58,64,67,12,16,25,37,28,40,48,59,30,41,50,60,52,62,65,68,7,13,19,31,20,32,43,53,21,33,44,54,45,55,63,66);(2,4)(6,13,16,14)(15,31,37,34)(17,32,40,35)(18,33,41,36)(38,53,59,56)(39,54,60,57)(42,55,62,58)(61,66,68,67)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=f,f^2=g,g^2=h,h^2=i,b^a=bc,c^a=ci,e^a=ei224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1953);