Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476707 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
570 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 570 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) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14,11,15,23,9)(7,16,19,26,18,12,20,31)(8,21,24,13,25,29,36,30)(17,33,35,32,34,28,22,27);(5,7,10,19,11,18,23,20)(6,12,9,26,15,16,14,31)(8,22,24,17,25,35,36,34)(13,28,21,32,30,33,29,27);(5,8,23,36,11,25,10,24)(6,13,14,29,15,30,9,21)(7,17,19,35,18,34,20,22)(12,27,26,28,16,32,31,33);(2,4)(5,9,10,15,11,14,23,6)(7,16,20,26,18,12,19,31)(8,21,24,30,25,29,36,13)(17,27,22,33,34,32,35,28)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=efi,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,h^b=hi,d^c=de,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386686);