Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7453 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1365 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| prime radical | C23 | #361 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
148 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 148 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:
(1,2,3,4)(5,6)(7,20,22,51)(8,26)(9,33)(10,35)(11,18)(12,17)(13,19)(14,21,39,54)(15,30)(16,29)(23,49,27,42)(24,58,36,60)(25,59,38,40)(28,48)(31,62)(32,61)(34,46)(37,43)(41,63,47,57)(44,65)(45,64)(50,66,67,55)(52,53,68,56);(5,7,9,23,28,56,65,55,12,25,31,21,10,36,34,63,13,22,8,27,29,53,64,66,11,38,32,54,37,24,30,57)(6,14,15,40,43,67,61,68,18,42,45,51,16,47,26,60,19,39,46,59,35,50,62,52,17,49,44,20,48,41,33,58)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,e^2=fg,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,g^b=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1559);