Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| 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 | |
650 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 178 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)(11,13)(12,19)(14,18)(15,22)(16,20)(17,23)(21,24)(25,26,31,35,29,34,30,39)(27,33,28,38,36,32,37,40);(1,3,6,2)(4,7,5,8)(9,11,13,10)(12,15,21,17)(14,18,20,16)(19,23,24,22)(25,27,30,37,29,36,31,28)(26,32,39,38,34,33,35,40);(1,4,6,5)(2,7,3,8)(9,12,20,23,13,21,14,22)(10,15,18,24,11,17,16,19)(25,28,31,36,29,37,30,27)(26,33,39,40,34,32,35,38);(1,5,6,4)(2,7,3,8)(10,11)(15,17)(16,18)(19,24)(27,36)(28,37)(32,33)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=ef,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,d^b=dh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 126635);