Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #126437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7800 |
| 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 | |
291 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 291 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:
(1,2,6,13,7,11,24,12)(3,14,17,26,18,32,23,31)(4,9,5,20,19,10,22,30)(8,25,28,16,29,21,27,15);(1,3,7,18)(2,8)(4,15,19,16)(5,21,22,25)(6,23,24,17)(9,26)(10,31)(11,29)(12,28)(13,27)(14,20)(30,32);(1,4,7,19)(2,9)(3,15,18,16)(5,6,22,24)(8,26)(10,11)(12,20)(13,30)(14,28)(17,21,23,25)(27,32)(29,31)Polycyclic presentation:
2^9|a^2=g,b^2=fg,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=df,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, 33633);