Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #404632 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2644 |
| 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 | |
701 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 701 conjugacy classes of subgroups — too many to draw. Only its 170 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)(5,6,12,16,11,17,10,27)(7,18,21,13,22,31,23,32)(8,14,15,9,25,29,28,26)(19,33,24,20,30,34,35,36);(5,7,10,23,11,22,12,21)(6,13,27,18,17,32,16,31)(8,24,28,19,25,35,15,30)(9,20,14,33,26,36,29,34);(5,8,10,15,11,25,12,28)(6,14,16,26,17,29,27,9)(7,19,21,35,22,30,23,24)(13,20,18,34,32,36,31,33);(2,4)(5,9,10,14,11,26,12,29)(6,15,16,25,17,28,27,8)(7,20,23,33,22,36,21,34)(13,30,31,35,32,19,18,24)Polycyclic presentation:
2^9|a^2=efi,b^2=ef,c^2=e,d^2=g,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311886);