Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #102 |
| commutator | C2×C8 | C2×C16 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | D4⋊C16 |
| prime radical | C23 | #350 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
124 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,19,15,21,36,51,14,20,35,50,33,48,62,70,16,22,37,52,38,53,65,72,31,46,61,69,12,18,32,47)(11,23,25,54,27,41,58,49,28,55,59,71,30,17,39,40,29,42,60,66,24,43,56,67,26,45,34,63,57,44,64,68);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,30)(13,34)(14,28)(15,24)(16,29)(18,45)(19,49)(20,43)(21,23)(22,44)(25,35)(26,31)(27,38)(32,58)(33,57)(36,39)(37,59)(40,50)(41,46)(42,53)(47,66)(48,55)(51,63)(52,67)(54,70)(56,62)(60,61)(64,65)(68,69)(71,72)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,f^2=i,g^2=h,h^2=i,b^a=bc,c^a=ce,c^b=cf,d^b=def,f^b=fi,g^b=gi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1124);