Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133406 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | C2×#120 | C2×C4 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
540 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18)(19,27)(20,31,33,24)(21,25)(22,26)(23,29)(28,40,43,47)(30,41)(32,38,46,37)(34,39)(35,42,45,48)(36,44);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,22,30)(18,23,26,36)(20,28,33,43)(21,29,34,44)(24,35,31,45)(25,27,39,41)(32,42,46,48)(37,47,38,40);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20,22,33)(18,24,26,31)(19,28,30,43)(21,32,34,46)(23,35,36,45)(25,38,39,37)(27,40,41,47)(29,42,44,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,e^b=ehi,f^b=fg,d^c=dh,f^c=fi,e^d=ei44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40857);