Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130730 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #61086 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
768 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 768 conjugacy classes of subgroups — too many to draw. Only its 98 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:
(1,2,3,6,4,5,7,8)(9,10,13,19,29,18,35,23,15,20,30,17,14,11,21,24)(12,22,25,39,26,16,31,33,27,36,40,38,28,34,37,32);(2,5)(3,7)(9,11)(10,14)(12,22)(13,19)(15,18)(16,28)(17,30)(20,29)(21,23)(24,35)(25,33)(26,34)(27,36)(31,39)(32,40)(37,38);(2,5)(3,7)(9,12,15,27)(10,16,20,34)(11,22,18,36)(13,25,30,40)(14,28,29,26)(17,32,19,33)(21,37,35,31)(23,38,24,39)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=hi,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,f^b=fgh,h^b=hi,e^c=ei,f^c=fgi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38245);