Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7213 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #342 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C2×#405 | C22 |
| socle | C22 | #1282 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×#405 ▷ C22×C4 ▷ C2 ▷ C1 | |
264 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 76 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)(9,10,13,19,16,22,35,20,15,21,34,49,37,50,14,36)(11,23,25,51,28,54,60,52,27,53,59,69,62,72,26,61)(12,18,29,43,32,46,30,63,31,45,33,48,64,71,65,44)(17,38,39,67,42,68,66,24,41,55,47,58,70,56,40,57);(1,2,4,7,5,8,3,6)(9,11,14,26,37,62,34,59,15,27,35,60,16,28,13,25)(10,17,20,40,50,70,19,47,21,41,36,66,22,42,49,39)(12,24,30,56,64,67,29,55,31,57,65,68,32,58,33,38)(18,23,44,52,71,72,48,51,45,53,63,61,46,54,43,69)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=fgh,h^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,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, 565);