Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13694 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| socle | C22 | C22×D16 |
| prime radical | C23 | C2×D16 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
330 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 256 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10,15,20)(11,21,26,47)(12,27,30,59)(13,31,33,61)(14,34,32,19)(16,35,38,23)(17,39,41,22)(18,42,43,28)(24,53,55,72)(25,56,54,46)(29,60,49,58)(36,66,67,71)(37,52,62,65)(40,50,57,68)(44,64,69,48)(45,63,70,51);(2,5)(6,8);(9,11,22,48,13,24,49,65,32,54,68,66,42,63,30,35,15,26,39,64,33,55,29,52,14,25,50,71,28,51,12,23)(10,16,27,45,18,36,57,56,34,62,60,72,61,69,41,47,20,38,59,70,43,67,40,46,19,37,58,53,31,44,17,21);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=e,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420264);