Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C4×C8 |
| Frattini, cosocle | C2×#408 | C22 |
| socle | C22 | #1316 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C2×#408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
378 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 378 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) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 68 points:
(5,6,9,15,11,16,30,48,12,17,31,49,10,32,35,47)(7,18,20,50,21,51,55,67,22,42,56,64,33,63,28,60)(8,23,25,57,26,14,41,43,27,44,29,61,34,45,59,68)(13,36,37,19,39,52,46,54,40,62,65,24,38,58,66,53);(1,2,3,4)(5,7)(6,13)(8,24)(9,28)(10,33)(11,21)(12,22)(14,42)(15,46)(16,39)(17,40)(18,45)(19,27)(20,30)(23,51)(25,36)(26,53)(29,62)(31,55)(32,38)(34,54)(35,56)(37,47)(41,52)(43,67)(44,63)(48,65)(49,66)(50,57)(58,59)(60,68)(61,64)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=g,f^2=i,g^2=i,b^a=bc,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=df,f^b=fi,h^b=hi,d^c=dhi,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1292);