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)(9,10,13,19,15,21,36,49,16,22,37,20,30,48,14,38)(11,23,25,50,27,52,59,72,28,53,60,51,54,71,26,61)(12,29,31,62,33,45,32,47,34,63,35,65,64,18,43,44)(17,39,40,67,41,68,66,24,42,55,46,57,69,56,70,58);(1,2,4,7,5,8,3,6)(9,11,14,26,30,54,37,60,16,28,36,59,15,27,13,25)(10,17,20,40,21,41,38,66,22,42,19,46,48,69,49,70)(12,24,32,56,64,67,31,55,34,58,43,68,33,57,35,39)(18,23,44,51,29,52,62,61,45,53,47,50,63,71,65,72)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,d^2=h,e^2=fgi,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, 524);