Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #126283 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1613 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| prime radical | C23 | #450 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
263 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 263 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,15,21,16,22,14,33)(11,23,27,34,28,40,26,36)(12,20,13,32,31,17,30,19)(18,35,37,25,38,24,39,29);(1,2,4,6)(3,8,7,5)(10,17)(12,13)(14,15)(18,34)(19,33)(20,22)(21,32)(23,37)(24,29)(25,35)(26,27)(30,31)(36,38)(39,40);(9,11,16,28)(10,18,22,38)(12,29,31,25)(13,24,30,35)(14,27,15,26)(17,34,20,36)(19,23,32,40)(21,39,33,37)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=i,d^2=f,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,h^a=hi,d^b=dfi,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi,h^c=hi,e^d=ei41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33642);