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:
(5,6,11,16)(7,14,19,34)(8,20)(9,25,18,12)(10,27,26,15)(13,29)(17,31)(21,33)(22,36)(23,28)(24,30)(32,35);(1,2)(3,4)(5,7,10,18,11,19,26,9)(6,12,15,14,16,25,27,34)(8,17,22,29,24,35,23,21)(13,28,31,30,33,36,32,20);(2,4)(5,8)(6,13)(7,17)(9,21)(10,22)(11,24)(12,28)(14,30)(15,31)(16,33)(18,29)(19,35)(20,34)(23,26)(25,36)(27,32);(5,9,26,19,11,18,10,7)(6,14,27,12,16,34,15,25)(8,21,23,35,24,29,22,17)(13,30,32,28,33,20,31,36)Polycyclic presentation:
2^9|a^2=eh,b^2=e,d^2=ei,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, 408085);