Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13694 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| socle | C22 | C22×D16 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
374 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(10,15)(11,16)(12,20)(13,23)(17,32)(18,35)(19,28)(21,37)(22,31)(24,36)(25,26)(27,33)(29,30)(34,40)(38,39);(2,5)(6,8);(9,10,16,26,12,17,29,35,23,33,37,40,36,38,22,28,14,19,31,39,24,34,21,27,13,18,30,32,20,25,11,15);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|c^2=fg,d^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420263);