Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7396 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C2×M16 | C2×C8 |
| Frattini, cosocle | #1729 | C22 |
| socle | C22 | #1343 |
| prime radical | C24 | #104 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ #1729 ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
234 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 48 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:
(2,5,6,7)(9,10,13,19,15,20,33,49,16,21,34,50,14,35,32,62)(11,22,24,51,25,44,56,69,26,52,57,66,36,65,63,72)(12,27,29,58,30,18,43,45,31,46,61,70,37,47,64,71)(17,38,39,54,41,60,68,23,42,53,48,28,40,59,67,55);(1,2)(3,7)(4,6)(5,8)(9,11,15,25,16,26,14,36)(10,17,35,40,21,42,20,41)(12,28,37,23,31,54,30,55)(13,24,33,56,34,57,32,63)(18,44,46,52,47,65,27,22)(19,48,62,68,50,39,49,67)(29,60,64,38,61,59,43,53)(45,69,70,66,71,72,58,51)Polycyclic presentation:
2^9|a^2=d,b^2=eh,c^2=fhi,d^2=g,e^2=i,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eh,c^b=cf,d^b=defhi,e^b=ei,h^b=hi,d^c=dhi,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1150);