Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C23 | C4×C16 |
| Frattini, cosocle | C22×M32 | C22 |
| socle | C22 | #1322 |
| prime radical | C24 | C2×C16 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C22×M32 ◁ G | |
| LECS | G ▷ C22×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
216 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 68 points:
(5,6,8,14,9,15,22,35,10,16,23,36,24,37,46,58,11,13,21,31,25,32,44,53,26,33,45,54,47,55,63,66)(7,12,18,27,19,29,41,48,20,30,42,49,43,52,62,64,17,28,38,34,39,50,59,56,40,51,60,57,61,65,68,67);(1,2,3,4)(5,7)(6,12,13,28)(8,18)(9,19)(10,20)(11,17)(14,34,31,27)(15,29,32,50)(16,30,33,51)(21,38)(22,41)(23,42)(24,43)(25,39)(26,40)(35,56,53,48)(36,57,54,49)(37,52,55,65)(44,59)(45,60)(46,62)(47,61)(58,67,66,64)(63,68)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=g,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fi,c^b=ci,d^b=df,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1262);