Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458126 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2571 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
548 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 170 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:
(1,2,4,5)(3,6,7,8)(9,10,15,19,14,20,13,16)(11,17,22,31,24,33,23,30)(12,25,29,18,26,34,28,36)(21,37,27,40,38,35,39,32);(2,5)(6,8)(10,16,20,19)(12,26)(17,31,33,30)(18,34,36,25)(21,38)(27,39)(28,29)(32,35,40,37);(1,3)(2,6)(4,7)(5,8)(9,11,13,23,14,24,15,22)(10,17,19,31,20,33,16,30)(12,27,28,21,26,39,29,38)(18,35,34,32,36,37,25,40);(3,7)(6,8)(9,12,15,29,14,26,13,28)(10,18,16,25,20,36,19,34)(11,21,23,39,24,38,22,27)(17,32,31,37,33,40,30,35)Polycyclic presentation:
2^9|a^2=efg,b^2=g,c^2=fh,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,f^c=fi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383277);