Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512939 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
351 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 351 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(10,15)(11,16)(12,20)(13,23)(17,25)(18,34)(19,35)(21,37)(22,31)(24,36)(26,27)(28,33)(29,30)(32,40)(38,39);(2,5)(6,8);(9,10,16,27,12,18,29,35,23,33,37,40,36,38,22,25,14,17,31,39,24,32,21,28,13,19,30,34,20,26,11,15);(1,3,4,7)(2,6,5,8)(10,17)(15,25)(18,32)(19,33)(26,38)(27,39)(28,35)(34,40)Polycyclic presentation:
2^9|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=hi,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72232014, #72232017, #72232019, … |
| maximal normal subgroup of | #72232014, #72232017, #72232019, … |
| maximal quotient of | #72321065, #72322759 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420276);