Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150414 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2128 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #407 | C23 |
| socle | C2 | #11772 |
| prime radical | #24285 | C2 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C8 ◁ #2929 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
432 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 54 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:
(2,8,10,24)(3,13)(4,18,21,22)(9,25,26,23)(11,28,12,19)(14,29,27,30)(15,31,20,32)(16,17);(1,2)(3,14)(4,19)(5,12)(6,11)(7,10)(8,18)(9,16)(13,27)(15,23)(17,26)(20,25)(21,28)(22,24)(29,31)(30,32);(1,3,6,17,7,13,5,16)(2,9,12,27,10,26,11,14)(4,20,18,31,21,15,22,32)(8,23,19,29,24,25,28,30)Polycyclic presentation:
2^9|a^2=e,c^2=g,d^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,c^b=cf,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,h^b=hi,d^c=dhi,e^d=ehi,f^d=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57658);