Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7225 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,ℤ/4) |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C2×#405 | C22 |
| socle | C22 | #1290 |
| prime radical | C24 | #108 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×#405 ▷ C22×C4 ▷ C2 ▷ C1 | |
238 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(2,6)(3,4)(7,8)(9,10,13,20,16,23,38,52,17,22,39,51,15,21,14,40)(11,24,26,53,29,56,62,72,30,55,63,70,28,54,27,64)(12,31,32,65,35,19,33,47,36,50,37,66,34,49,67,48)(18,41,42,59,46,69,68,25,45,57,43,60,44,58,71,61);(1,2,4,7,5,8,3,6)(9,11,14,27,15,28,39,63,17,30,38,62,16,29,13,26)(10,18,20,43,23,46,52,71,22,45,51,42,21,44,40,68)(12,25,33,58,34,59,32,57,36,61,67,69,35,60,37,41)(19,24,48,53,50,56,65,72,49,55,47,70,31,54,66,64)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,d^2=h,e^2=fg,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgi,g^a=gi,d^b=dg,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 523);