Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #126387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7795 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| prime radical | #31231 | C2 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
283 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 283 conjugacy classes of subgroups — too many to draw. Only its 52 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:
(1,2,8,12,6,13,7,24)(3,14,18,26,19,32,17,30)(4,11,5,22,21,10,20,29)(9,25,27,16,28,23,31,15);(1,3)(2,9)(4,15)(5,23)(6,19)(7,18)(8,17)(10,26)(11,30)(12,31)(13,28)(14,29)(16,21)(20,25)(22,32)(24,27);(1,4)(2,10,13,11)(3,15)(5,7)(6,21)(8,20)(9,26,28,30)(12,22,24,29)(14,31,32,27)(16,19)(17,25)(18,23)Polycyclic presentation:
2^9|a^2=g,c^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=dfi,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,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, 33583);