Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ #1744 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
670 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 670 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,7)(5,8)(9,10,13,16)(11,18,20,15)(12,21,23,24)(14,19,22,17)(25,26,30,35,29,33,39,34)(27,32,37,40,36,31,28,38);(1,3,6,2)(4,7,5,8)(9,11,13,20)(10,15,16,18)(12,17,23,19)(14,21,22,24)(25,27,30,37,29,36,39,28)(26,31,35,38,33,32,34,40);(1,4,6,5)(2,7,3,8)(9,12,22,18,13,23,14,15)(10,11,19,24,16,20,17,21)(25,28,39,36,29,37,30,27)(26,32,35,40,33,31,34,38);(1,5,6,4)(2,7,3,8)(10,16)(11,20)(17,19)(21,24)(26,33)(27,36)(28,37)(34,35)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=ef,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cei,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 126609);