Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153213 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2226 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C23 | #509 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
174 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,17,13,18,25,15)(11,20,22,30,23,16,24,32)(14,19,26,33,27,34,35,28)(21,38,36,31,39,29,40,37);(1,3,4,7)(2,5,6,8)(10,15)(11,21,23,39)(12,25)(14,28)(16,31,20,37)(17,18)(19,35)(22,40,24,36)(26,34)(27,33)(29,32,38,30);(2,6)(5,8)(9,11,12,22,13,23,25,24)(10,16,17,32,18,20,15,30)(14,29,26,37,27,38,35,31)(19,36,33,39,34,40,28,21)Polycyclic presentation:
2^9|a^2=d,b^2=deh,c^2=d,d^2=i,f^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,f^b=fg,g^b=gh,h^b=hi,f^c=fg,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60612);