Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417558 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2008 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 2008 conjugacy classes of subgroups — too many to draw. Only its 210 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)(3,7,8,5)(9,10)(11,16)(12,23,26,35)(13,18)(14,19)(15,20)(17,25,33,28)(21,37,24,40)(22,31)(27,34)(29,36)(30,38,32,39);(1,3)(2,5)(4,8)(6,7)(9,11,15,13)(10,16,20,18)(12,21,26,24)(14,27,29,22)(17,32,33,30)(19,34,36,31)(23,37,35,40)(25,39,28,38);(9,12)(10,17)(11,21)(13,24)(14,28)(15,26)(16,30)(18,32)(19,35)(20,33)(22,39)(23,36)(25,29)(27,38)(31,40)(34,37);(3,8)(5,7)(9,13,15,11)(10,18,20,16)(12,24,26,21)(14,22,29,27)(17,32,33,30)(19,31,36,34)(23,37,35,40)(25,38,28,39)Polycyclic presentation:
2^9|a^2=eh,b^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356726);