Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#102 |
| prime radical | #466 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
283 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,11,7,12,6,10)(3,13,14,16,15,4,9,8)(17,18)(19,28)(20,33)(21,36)(22,38)(23,27)(24,30)(25,35)(26,34)(29,42)(31,39)(32,48)(37,41)(40,47)(43,46)(44,45);(1,3,6,9,7,15,5,14)(2,8,10,4,12,16,11,13)(17,19,21,31,34,45,35,46,23,29,20,30,22,32,37,47)(18,24,25,28,38,43,36,48,27,39,41,42,26,40,33,44);(1,4,5,8,7,13,6,16)(2,9,11,3,12,14,10,15)(19,29)(24,39)(28,42)(30,31)(32,45)(40,43)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,c^b=ci,d^b=dh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36200);