Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×#124 | C23 |
| socle | C22 | C2×#353 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
656 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 656 conjugacy classes of subgroups — too many to draw. Only its 130 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:
(5,6)(7,18,20,33)(8,14)(9,25)(10,27,11,16)(12,17)(13,19,29,34)(15,23,32,24)(21,28,22,31)(26,36,35,30);(1,2,3,4)(5,7,10,20,12,22,11,21)(6,13,16,29,17,30,27,36)(8,19,23,34,9,26,24,35)(14,28,15,31,25,33,32,18);(5,8,12,9)(6,14,17,25)(7,19,22,26)(10,23,11,24)(13,28,30,33)(15,27,32,16)(18,29,31,36)(20,34,21,35)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=h,f^2=i,g^2=i,b^a=bd,d^a=dg,f^a=fhi,g^a=gh,h^a=hi,d^b=di,e^b=eghi,e^d=ehi,f^e=fi,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59062);