Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149901 |
| 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×Q8 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | #24223 | C2 |
| LCS | G ▷ C22×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
488 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(2,4)(5,6)(7,15,19,31)(8,21,16,32)(9,24,26,13)(10,14)(11,18,29,34)(12,23)(17,33,20,36)(22,28,35,30)(25,27);(5,7,16,34)(6,11,27,36,14,28,12,31)(8,22,10,17)(9,20,25,18)(13,29,32,33,24,30,21,15)(19,23,35,26);(1,2,3,4)(7,17)(11,28)(15,33)(18,35)(19,20)(22,34)(29,30)(31,36)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,c^2=e,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ci,d^b=dg,f^b=fg,g^b=gh,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60100);