Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500887 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3050 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
781 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 781 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) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,20)(8,24,27,34)(9,13)(10,18)(11,16)(12,19)(14,25,31,36)(15,23)(17,22)(21,33,35,30)(26,32,28,29);(1,2)(3,4)(5,7,11,22,12,23,10,9)(6,13,18,15,19,17,16,20)(8,21,28,36,27,35,26,25)(14,29,33,24,31,32,30,34);(2,4)(5,8)(6,14)(7,21)(9,25)(10,26)(11,28)(12,27)(13,29)(15,24)(16,30)(17,32)(18,33)(19,31)(20,34)(22,36)(23,35);(5,9,10,23,12,22,11,7)(6,15,16,13,19,20,18,17)(8,25,26,35,27,36,28,21)(14,24,30,29,31,34,33,32)Polycyclic presentation:
2^9|a^2=eh,b^2=ei,d^2=e,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408084);