Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11502 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator, prime radical | C24 | C22×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | C2×#324 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
392 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,8,13,9,14,19,27,10,12,18,24,20,25,31,34)(7,11,16,22,17,23,30,33,15,21,28,26,29,32,36,35);(1,2,3,4)(7,15)(11,21)(16,28)(17,29)(22,26)(23,32)(30,36)(33,35);(5,7)(6,11,12,21)(8,16)(9,17)(10,15)(13,26,24,22)(14,23,25,32)(18,28)(19,30)(20,29)(27,35,34,33)(31,36)Polycyclic presentation:
2^9|a^2=f,b^2=d,c^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ci,e^c=ei,f^c=fg,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41463);