Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13697 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D16 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C22×D32 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
314 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 94 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 — 3 generators on 40 points:
(2,4)(5,6,7,8)(10,16)(11,20)(12,17)(13,24)(14,27)(18,35)(19,36)(21,26)(22,39)(23,31)(25,38)(28,40)(29,30)(32,37)(33,34);(1,2,3,4)(6,8)(9,10,17,30,13,18,33,36,27,23,38,39,40,37,26,20,15,11,21,32,28,22,25,31,14,19,34,35,24,29,12,16);(9,11,21,30,13,22,25,36,27,19,34,39,40,29,12,20,15,10,17,32,28,18,33,31,14,23,38,35,24,37,26,16)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=egi,d^2=gh,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,h^a=hi,c^b=ci92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60169);