Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509065 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
882 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 882 conjugacy classes of subgroups — too many to draw. Only its 146 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:
(1,2,3,4)(9,10,13,19,16,11,22,18,14,20,30,36,15,21,29,35)(12,24,26,40,31,25,37,39,27,17,32,34,28,23,38,33);(5,6)(7,8)(9,11)(10,16)(12,25)(13,19)(14,21)(15,20)(17,28)(18,29)(22,35)(23,27)(24,31)(26,40)(30,36)(32,34)(33,37)(38,39);(2,4)(6,8)(9,12,14,27)(10,17,20,24)(11,23,21,25)(13,26,30,32)(15,28,16,31)(18,33,35,39)(19,34,36,40)(22,37,29,38);(2,4)(10,18)(11,19)(13,29)(15,16)(17,33)(20,35)(21,36)(22,30)(23,34)(24,39)(25,40)(26,38)(28,31)(32,37)Polycyclic presentation:
2^9|a^2=efi,c^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417032);