Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #239311 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | #41 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1150 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 1150 conjugacy classes of subgroups — too many to draw. Only its 188 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) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,18)(11,19,22,34)(12,23,25,16)(13,27,29,17)(15,28,31,21)(20,36,24,35)(26,33,40,38)(30,37,32,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,22)(10,15,18,31)(12,24,25,20)(13,28,29,21)(16,32,23,30)(17,34,27,19)(26,37,40,39)(33,35,38,36);(5,8)(9,12,13,26,14,25,29,40)(10,16,17,33,18,23,27,38)(11,20,21,37,22,24,28,39)(15,30,19,35,31,32,34,36);(12,25)(16,23)(20,24)(26,40)(30,32)(33,38)(35,36)(37,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=e,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,d^c=dh,f^c=fh,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146509);