Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #117 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1797 |
| prime radical | C24 | C4⋊C8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
317 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 317 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,11,7,10,13,16,6,9,14,8,4,15,3,12)(17,18)(19,27)(20,31,33,24)(21,34)(22,26)(23,29)(25,35)(28,41,43,39)(30,40)(32,38,47,46)(36,42,45,48)(37,44);(1,3,4,14,6,13,7,5)(2,8,10,12,9,11,15,16)(17,19,21,29,22,30,35,44)(18,23,25,27,26,37,34,40)(20,28,32,42,33,43,47,48)(24,36,38,39,31,45,46,41);(2,9)(3,13)(5,14)(10,15)(17,20)(18,24)(19,28)(21,32)(22,33)(23,36)(25,38)(26,31)(27,39)(29,42)(30,43)(34,46)(35,47)(37,45)(40,41)(44,48)Polycyclic presentation:
2^9|a^2=f,b^2=d,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ci,f^c=fg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41900);