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 | |
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)(3,11,13,23,15,27,14,30)(4,8,18,24)(5,20)(6,9)(7,12,22,29,21,28,26,16)(10,17)(19,32,31,25);(1,3)(2,7)(4,16)(5,13)(6,15)(8,23)(9,21)(10,26)(11,25)(12,19)(14,17)(18,29)(20,22)(24,30)(27,32)(28,31);(7,21)(12,28)(16,29)(22,26)Polycyclic presentation:
2^9|a^2=e,d^2=gi,e^2=hi,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, 57647);