Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476659 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
526 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 526 conjugacy classes of subgroups — too many to draw. Only its 154 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:
(1,2,4,6)(3,5,7,8)(9,10,14,31,15,20,22,19)(11,21,32,34,26,38,25,16)(12,27,13,18,28,17,30,37)(23,39,40,33,24,36,29,35);(3,7)(5,8)(9,11,22,25,15,26,14,32)(10,16,19,38,20,34,31,21)(12,24,13,29,28,23,30,40)(17,35,37,39,27,33,18,36);(1,3,4,7)(2,5,6,8)(9,12,15,28)(10,17,20,27)(11,23,26,24)(13,14,30,22)(16,33,34,35)(18,19,37,31)(21,39,38,36)(25,40,32,29);(9,13,14,28,15,30,22,12)(10,18,19,27,20,37,31,17)(11,24,25,40,26,23,32,29)(16,33,21,36,34,35,38,39)Polycyclic presentation:
2^9|a^2=efgi,b^2=efh,c^2=g,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,e^c=ei,h^c=hi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387080);