Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233501 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
658 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 658 conjugacy classes of subgroups — too many to draw. Only its 162 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)(3,6,7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,34,31,28,29,33)(27,35,39,32,37,36,38,40);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,21)(23,24)(25,27,31,37)(26,32,28,40)(29,39,30,38)(33,36,34,35);(11,18)(13,21)(14,22)(16,24)(17,23)(19,20)(25,28,31,26)(27,36,37,35)(29,33,30,34)(32,38,40,39);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,24)(22,23)(27,37)(32,40)(35,36)(38,39)Polycyclic presentation:
2^9|a^2=ei,b^2=h,c^2=h,d^2=efi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151593);