Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143203 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #28246 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
451 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 451 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 3 generators on 36 points:
(5,6,12,18)(7,19,22,33)(8,23,27,15)(9,28,26,14)(10,17,11,16)(13,25,31,20)(21,35,29,32)(24,34,36,30);(1,2)(3,4)(5,7)(6,13)(8,24)(9,25)(10,29)(11,21)(12,22)(14,32)(15,33)(16,34)(17,30)(18,31)(19,23)(20,26)(27,36)(28,35);(2,4)(5,7)(6,13)(8,25)(9,24)(10,21)(11,29)(12,22)(14,33)(15,32)(16,30)(17,34)(18,31)(19,28)(20,27)(23,35)(26,36)Polycyclic presentation:
2^9|a^2=i,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,g^b=gi,d^c=di,e^c=eh,h^c=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72226862, #72226869, #72226872 |
| maximal normal subgroup of | #72226862, #72226869, #72226872 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50888);