Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509081 |
| 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 | |
879 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 879 conjugacy classes of subgroups — too many to draw. Only its 144 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)(9,10,16,24,17,26,30,25,15,21,13,20,14,11,22,19)(12,18,31,37,32,39,40,38,29,35,27,34,28,23,36,33);(5,6)(7,8)(9,11,15,26)(10,17,21,14)(12,23,29,39)(13,24,16,20)(18,32,35,28)(19,30,25,22)(27,37,31,34)(33,40,38,36);(2,4)(6,8)(9,12,15,29)(10,18,21,35)(11,23,26,39)(13,27,16,31)(14,28,17,32)(19,33,25,38)(20,34,24,37)(22,36,30,40);(2,4)(10,19)(11,24)(13,30)(14,17)(16,22)(18,33)(20,26)(21,25)(23,37)(27,40)(28,32)(31,36)(34,39)(35,38)Polycyclic presentation:
2^9|a^2=efi,b^2=i,c^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417041);