Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501090 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1214 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1214 conjugacy classes of subgroups — too many to draw. Only its 142 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)(7,17,9,22)(8,13)(10,25,26,15)(11,16)(12,19)(14,24)(18,33,23,32)(20,31,36,28)(21,29)(27,34)(30,35);(1,2)(3,4)(5,7,11,9)(6,12,16,14)(8,18,21,23)(10,24,26,19)(13,27,29,30)(15,22,25,17)(20,35,36,34)(28,32,31,33);(2,4)(5,8,11,21)(6,13,16,29)(7,18,9,23)(10,20,26,36)(12,27,14,30)(15,31,25,28)(17,32,22,33)(19,34,24,35);(5,9,11,7)(6,14,16,12)(8,18,21,23)(10,19,26,24)(13,27,29,30)(15,17,25,22)(20,35,36,34)(28,32,31,33)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fhi,d^c=di,e^c=eh,e^d=ei,f^d=fhi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72286145, #72286158, #72317548, … |
| maximal normal subgroup of | #72286145, #72286158, #72317548, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408291);