Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #333271 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #2537 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
580 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 580 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,4)(5,7)(6,8)(9,10,14,16,12,18,13,17)(11,19,22,24,20,15,21,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,38)(36,40)(37,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19)(25,27)(26,30)(28,35)(29,34)(31,33)(32,40)(36,37)(38,39);(1,4,5,6)(2,3,7,8)(10,16)(13,14)(15,23)(17,18)(19,24)(21,22)(26,31)(27,34)(29,37)(30,38)(33,40)(35,36);(2,7)(3,8)(10,17)(11,20)(13,14)(15,23)(16,18)(19,24)(27,35)(30,33)(34,36)(38,40)Polycyclic presentation:
2^9|a^2=fhi,c^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 250692);