Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1036 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 234 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)(3,4)(5,6,7,8)(9,10)(11,19)(12,18)(13,27)(15,16)(17,22)(20,32)(21,33)(23,36)(25,26)(28,30)(31,35)(34,40)(37,39);(9,11)(10,17)(12,20)(13,28)(14,19)(15,30)(16,21)(18,31)(22,27)(23,37)(24,32)(25,39)(26,34)(29,33)(35,36)(38,40);(2,4)(9,12,16,26)(10,18,15,25)(11,20,21,34)(13,23,27,36)(14,24,29,38)(17,31,30,39)(19,32,33,40)(22,35,28,37);(9,13)(10,14)(11,21)(12,23)(15,29)(16,27)(18,24)(19,28)(20,34)(22,33)(25,38)(26,36)(32,37)(35,40)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404630);