Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D16 |
| commutator | C16 | C4×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle, prime radical | C23 | #366 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2)(3,4)(5,6,7,9,8,10,11,12)(14,20)(15,16)(17,25)(18,31)(21,33)(22,37)(23,36)(24,35)(26,42)(27,30)(28,29)(32,41)(34,39)(38,43)(40,44);(2,4)(13,14,16,21,17,22,28,36,31,39,42,44,41,43,27,35,19,24,30,38,32,40,26,34,18,23,29,37,25,33,15,20)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=f,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efh,f^a=fh,h^a=hi152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1217);