Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150449 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11761 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #423 | C23 |
| prime radical | #2010 | C4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #2920 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
282 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 282 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:
(1,2,4,8,6,10,18,11)(3,12,15,30)(5,9,20,28,21,29,17,26)(7,22,24,19)(13,27,16,32)(14,25,31,23);(1,3)(2,7)(4,13)(5,19)(6,15)(8,25)(9,27)(10,24)(11,23)(12,28)(14,17)(16,18)(20,31)(21,22)(26,30)(29,32);(1,4,6,18)(2,8,10,11)(3,13,15,16)(5,17,21,20)(7,23,24,25)(9,26,29,28)(12,27,30,32)(14,19,31,22)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=gi,e^2=h,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,c^b=cf,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=dhi,e^d=eh,f^d=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57646);