Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D8 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×D8 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
307 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,16,12,8,24,15,13,17,9,25,11,26,29,28)(7,18,21,32,22,35,20,14,23,31,27,19,30,34,36,33);(2,4)(5,7,10,21,12,22,24,20,13,23,9,27,11,30,29,36)(6,14,16,31,8,19,15,34,17,33,25,18,26,32,28,35);(5,8)(6,11)(7,19)(9,28)(10,15)(12,17)(13,26)(14,30)(16,29)(18,20)(21,34)(22,33)(23,32)(24,25)(27,35)(31,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=gi,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dg,e^b=eg,g^b=gi,d^c=dg,e^c=eg,g^c=gi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43086);