Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150243 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#2215 | C2 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
339 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,7,8,5)(4,6)(9,10)(11,17)(12,16)(13,27)(14,29)(15,20)(18,23)(19,24)(21,30)(22,36)(25,33)(26,32)(28,40)(31,37)(34,38)(35,39);(1,3)(2,5)(4,8)(6,7)(10,16)(11,14)(12,23)(13,24)(17,33)(18,19)(20,32)(21,37)(22,28)(25,38)(26,39)(27,31)(29,30)(34,40)(35,36);(9,11,15,14)(10,17,20,29)(12,21,26,25)(13,22,28,24)(16,30,32,33)(18,34,35,31)(19,27,36,40)(23,38,39,37)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=gh,f^2=hi,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gh,h^a=hi,c^b=ci,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57440);