Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149792 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2126 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #423 | C23 |
| socle | C2 | #11507 |
| prime radical | #3444 | C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C42 ◁ C2×C42 ◁ #2920 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
352 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 56 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:
(1,2,4,10,5,9,20,27,6,11,18,29,21,8,19,26)(3,12,14,28,16,7,22,24,13,23,30,32,15,25,17,31);(1,3)(2,7)(4,17)(5,16)(6,13)(8,12)(9,23)(10,28)(11,25)(14,20)(15,21)(18,22)(19,30)(24,27)(26,31)(29,32);(2,8,11,9)(3,13)(4,18)(7,23,25,12)(10,27,29,26)(15,16)(19,20)(24,28,31,32)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=h,e^2=g,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dh,e^b=efhi,f^b=fi,e^c=ei,e^d=ehi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56988);