Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150414 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2128 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #407 | C23 |
| socle | C2 | #11754 |
| prime radical | #2010 | C22 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C42 ◁ #2920 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
283 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 283 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:
(1,2,4,10,6,11,18,8)(3,12,15,30,13,31,16,28)(5,19,17,29,22,27,21,9)(7,23,25,14,24,20,26,32);(1,3)(2,7)(4,15)(5,20)(6,13)(8,25)(9,28)(10,26)(11,24)(12,19)(14,17)(16,18)(21,32)(22,23)(27,31)(29,30);(1,4,6,18)(2,8,11,10)(3,13)(5,17,22,21)(7,24)(9,27,29,19)(15,16)(25,26)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=gi,e^2=i,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, 57612);