Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
726 conjugacy classes of subgroups; only the 314 normal subgroups are shown.
This group has 726 conjugacy classes of subgroups — too many to draw. Only its 314 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)(9,10)(11,20)(12,16)(13,18)(14,23)(15,19)(17,22)(21,24)(25,26,28,31,29,32,35,38)(27,30,34,37,33,36,39,40);(2,5)(6,8)(9,11,14,12,15,22,13,21)(10,16,18,20,19,24,23,17)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(1,3,4,7)(2,6,5,8)(9,12,14,22,15,21,13,11)(10,17,18,16,19,20,23,24)(27,33)(30,36)(34,39)(37,40);(9,13,15,14)(10,18,19,23)(11,21,22,12)(16,20,24,17)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=chi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 98370);