Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23444 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#120 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1206 conjugacy classes of subgroups; only the 298 normal subgroups are shown.
This group has 1206 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:
(1,2)(3,8)(4,6)(5,7)(9,10,13,18,14,17,24,30)(11,15,21,27,20,25,33,36)(12,16,23,29,22,28,34,38)(19,26,32,37,31,35,39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,26)(17,25)(18,27)(22,31)(23,32)(24,33)(28,35)(29,37)(30,36)(34,39)(38,40);(12,22)(15,25)(16,28)(19,31)(23,34)(27,36)(29,38)(32,39);(1,4,5,3)(2,6,7,8)(9,12,13,23,14,22,24,34)(10,16,18,29,17,28,30,38)(11,19,21,32,20,31,33,39)(15,26,27,37,25,35,36,40)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=eh,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,f^b=fi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266371);