Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190022 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1314 conjugacy classes of subgroups; only the 306 normal subgroups are shown.
This group has 1314 conjugacy classes of subgroups — too many to draw. Only its 306 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:
(1,2)(3,5)(4,6)(7,8)(9,10,14,20,15,19,12,17)(11,21,22,34,24,16,29,30)(13,26,28,32,27,18,25,33)(23,31,36,39,37,35,38,40);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,22)(13,23)(14,29)(15,24)(17,30)(18,31)(19,21)(20,34)(25,36)(26,35)(27,37)(28,38)(32,39)(33,40);(2,6)(5,8)(9,12,15,14)(10,17,19,20)(11,22,24,29)(13,25,27,28)(16,30,21,34)(18,32,26,33)(23,36,37,38)(31,39,35,40);(9,13)(10,18,19,26)(11,23)(12,25)(14,28)(15,27)(16,31,21,35)(17,32,20,33)(22,36)(24,37)(29,38)(30,39,34,40)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^b=hi,g^d=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299567);