Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244791 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D42 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
546 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 546 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 512 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,12,16,14,18,13,17)(11,19,20,24,22,23,21,15)(25,26,30,29)(27,31,34,37)(28,32,36,38)(33,39,40,35);(2,5)(6,7)(9,11,14,22)(10,15,18,24)(12,21,13,20)(16,23,17,19)(25,27,30,34)(26,31,29,37)(28,35,36,39)(32,33,38,40);(10,16)(12,13)(15,23)(17,18)(19,24)(20,21)(25,28,30,36)(26,32,29,38)(27,33,34,40)(31,39,37,35);(1,3,4,8)(2,6,5,7)(10,17)(12,13)(15,19)(16,18)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=ei,b^2=hi,c^2=i,d^2=ef,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi,e^d=eh,f^d=fh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 219354);