Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149895 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×AΣL(1,𝔽16) |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | #23908 | C2 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
640 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 640 conjugacy classes of subgroups — too many to draw. Only its 68 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)(7,11,17,24)(8,12)(9,14,22,13)(10,15)(16,23,30,34)(18,25,21,29)(19,27,33,26)(20,28)(31,35,32,36);(5,7,10,18)(6,11,14,24,15,25,13,29)(8,16,20,31)(9,21,22,17)(12,23,27,34,28,35,26,36)(19,32,33,30);(2,4)(5,8,10,20)(6,12,15,28)(7,16,18,31)(9,19,22,33)(11,23,25,35)(13,26,14,27)(17,30,21,32)(24,34,29,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dgi,f^b=fgi,g^b=gh,f^d=fhi,g^d=gi,g^f=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59855);