Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150123 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C23 | C2×SD32 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
316 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 256 |
A minimal faithful permutation representation — 3 generators on 68 points:
(2,4)(5,6,9,15,32,48,65,45,11,17,30,14,36,44,56,53,12,18,34,46,28,43,29,62,20,51,8,26,10,16,33,49)(7,19,22,39,35,47,27,13,24,38,21,50,58,61,63,40,25,55,59,67,57,68,64,41,37,66,60,54,23,42,31,52);(5,7,20,24,12,25,11,37)(6,13,17,52,18,41,51,40)(8,27,34,63,30,64,9,31)(10,35,28,58,36,57,32,23)(14,42,46,68,26,61,15,47)(16,50,48,39,44,54,43,67)(19,53,66,62,55,49,38,45)(21,56,59,65,60,33,22,29);(1,2,3,4)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=e,d^2=ghi,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,f^b=fg,g^b=gh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60185);