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 | |
303 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 303 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)(3,11)(4,16,18,27)(5,10)(6,8)(7,13)(9,20,24,21)(12,29)(14,26)(15,30)(17,22)(19,25)(23,32)(28,31);(1,3)(2,7)(4,17)(5,12)(6,15)(8,23)(9,26)(10,28)(11,27)(13,21)(14,19)(16,30)(18,31)(20,32)(22,25)(24,29);(2,8)(3,12,15,14)(4,18)(7,22,23,28)(10,25)(11,26,30,29)(13,31,32,17)(20,21)Polycyclic presentation:
2^9|a^2=e,c^2=h,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, 57610);