Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393921 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62845 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2554 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 2554 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(9,10,15,20)(11,21,25,31)(12,26,29,33)(13,19,14,18)(16,24,32,23)(17,27,34,28)(22,36,38,30)(35,39,40,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,22)(13,24)(14,23)(15,25)(17,30)(18,31)(19,21)(20,32)(26,35)(27,39)(28,37)(29,38)(33,40)(34,36);(2,5)(6,8)(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,33)(19,26)(20,34)(21,35)(23,37)(24,39)(25,38)(31,40)(32,36);(1,3,4,7)(2,6,5,8)(10,18)(13,14)(16,31)(17,33)(19,20)(21,32)(23,24)(26,34)(27,28)(30,40)(35,36)(37,39)Polycyclic presentation:
2^9|a^2=i,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311126);