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)(9,10,16,24,17,26,32,25,14,21,13,20,15,11,22,19)(12,27,33,36,34,23,38,35,30,18,29,39,31,28,37,40);(5,6)(7,8)(9,11,14,26)(10,17,21,15)(12,28,30,23)(13,24,16,20)(18,31,27,34)(19,32,25,22)(29,36,33,39)(35,37,40,38);(2,4)(6,8)(9,12,14,30)(10,18,21,27)(11,23,26,28)(13,29,16,33)(15,31,17,34)(19,35,25,40)(20,36,24,39)(22,37,32,38);(2,4)(10,19)(11,24)(13,32)(15,17)(16,22)(18,35)(20,26)(21,25)(23,39)(27,40)(28,36)(29,38)(31,34)(33,37)Polycyclic presentation:
2^9|a^2=ef,b^2=i,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, 417033);