Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
392 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 256 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6,9,12)(7,13,14,20)(8,16,17,11)(10,18,19,15)(22,28)(23,24)(25,26)(29,31)(30,33)(32,36)(34,35);(5,7)(6,10,12,19)(8,15)(9,14)(11,13,16,20)(17,18)(21,22,24,30,26,32,35,36,27,33,34,28,25,31,23,29);(1,3,4,2)(7,14)(10,19)(13,20)(15,18)(21,23,25,34,27,35,26,24)(22,29,31,28,33,36,32,30)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=eg,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fg,g^a=gi,h^a=hi,c^b=cgh,d^b=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72318619, #72318636, #72318675, … |
| maximal normal subgroup of | #72318619, #72318636, #72318675, … |
| maximal quotient of | #72322404, #72322472 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35813);