Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150157 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#2246 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
376 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 376 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)(5,6,9,15,12,17,11,31)(7,18,20,26,14,33,29,35)(8,22,25,36,28,21,27,32)(10,19,24,34,30,23,13,16);(5,7,18,31,10,21,8,23,9,29,26,17,24,36,25,34,12,14,33,15,30,22,28,19,11,20,35,6,13,32,27,16);(1,2,3,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=e,d^2=ghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=di,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216288, #72216315, #72216324, … |
| maximal normal subgroup of | #72216288, #72216315, #72216324, … |
| maximal quotient of | #72216283, #72216288 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60095);