Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #126349 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7791 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
264 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 264 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11,7,9,5,22)(3,12,14,27,16,30,15,29)(4,17,19,26,20,28,21,10)(8,18,25,13,23,31,24,32);(1,3)(2,8,9,23)(4,18)(5,14)(6,15)(7,16)(10,27,26,29)(11,24,22,25)(12,28,30,17)(13,21)(19,32)(20,31);(2,9)(4,19)(5,6)(8,23)(10,28)(12,29)(13,31)(14,15)(17,26)(18,32)(20,21)(27,30)Polycyclic presentation:
2^9|a^2=g,b^2=fg,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,d^b=df,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi,e^d=ei38 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33545);