Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393921 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2318 conjugacy classes of subgroups; only the 298 normal subgroups are shown.
This group has 2318 conjugacy classes of subgroups — too many to draw. Only its 298 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) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(9,10,13,18,15,19,14,20)(11,21,23,32,24,31,25,16)(12,26,27,33,28,17,29,34)(22,35,36,30,37,40,38,39);(1,2,4,6)(3,7,8,5)(9,11)(10,16)(12,22)(13,25)(14,23)(15,24)(17,30)(18,31)(19,32)(20,21)(26,39)(27,38)(28,37)(29,36)(33,40)(34,35);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,22)(13,29)(14,27)(15,28)(16,30)(18,33)(19,26)(20,34)(21,35)(23,38)(24,37)(25,36)(31,40)(32,39);(10,18)(13,14)(16,31)(17,33)(19,20)(21,32)(23,25)(26,34)(27,29)(30,40)(35,39)(36,38)Polycyclic presentation:
2^9|a^2=e,b^2=g,c^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310586);