Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153125 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2199 |
| commutator, Frattini, cosocle | #532 | C23 |
| socle | C2 | #13348 |
| prime radical | #33567 | C2 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
511 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 511 conjugacy classes of subgroups — too many to draw. Only its 50 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)(3,12,16,24)(4,9)(5,18)(6,21)(7,11)(8,14,25,15)(10,17)(13,28,20,23)(19,26)(22,27)(29,31,32,30);(3,13)(5,19)(8,23)(10,27)(12,29)(14,31)(15,30)(16,20)(17,22)(18,26)(24,32)(25,28);(1,3)(2,8)(4,15)(5,20)(6,14)(7,16)(9,24)(10,28)(11,25)(12,21)(13,19)(17,31)(18,32)(22,30)(23,27)(26,29)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fgh,g^a=gi,c^b=cf,d^b=dh,f^b=fi,g^b=gh,d^c=dg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fh,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60380);