Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458126 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2537 |
| 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 | |
708 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 708 conjugacy classes of subgroups — too many to draw. Only its 202 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)(11,17)(12,26)(13,32)(14,21)(15,20)(16,22)(18,29)(19,25)(23,35)(24,34)(27,36)(28,33)(30,37)(31,38)(39,40);(2,5)(6,8)(9,11,15,13,16,25,14,24)(10,17,20,32,22,19,21,34)(12,27,30,39,31,28,29,23)(18,35,26,36,37,40,38,33);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,23)(13,28)(14,30)(15,29)(16,31)(17,33)(19,36)(20,38)(21,26)(22,37)(24,27)(25,39)(32,40)(34,35);(3,7)(6,8)(9,13,15,25,16,24,14,11)(10,19,21,32,22,17,20,34)(12,28,29,39,31,27,30,23)(18,36,26,40,37,33,38,35)Polycyclic presentation:
2^9|a^2=e,b^2=g,c^2=e,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=di,d^c=de,f^c=fi,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383050);