Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152145 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12708 |
| commutator | #117 | C22×C4 |
| Frattini, cosocle | #535 | C23 |
| prime radical | #61945 | C2 |
| LCS | G ▷ #117 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #3508 ◁ G | |
| LECS | G ▷ #535 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
567 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 567 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(1,2,6,12,7,14,5,11)(3,15,18,31,20,28,32,26)(4,10,21,30,22,13,24,9)(8,23,27,19,29,17,25,16);(1,3)(2,8)(4,16)(5,23)(6,17)(7,20)(9,25)(10,28)(11,26)(12,31)(13,15)(14,29)(18,21)(19,22)(24,32)(27,30);(1,4,7,22)(2,9)(3,16,20,19)(5,21,6,24)(8,25)(10,11)(12,13)(14,30)(15,31)(17,32,23,18)(26,28)(27,29)Polycyclic presentation:
2^9|a^2=f,c^2=hi,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,f^c=fi,e^d=ei,f^d=fhi,g^f=gi35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59349);