Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4×C8 | C23 |
| commutator | C22 | C2×C82 |
| socle | C23 | C2×C4×C8 |
| prime radical | C24 | C4×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
536 conjugacy classes of subgroups; only the 254 normal subgroups are shown.
This group has 536 conjugacy classes of subgroups — too many to draw. Only its 254 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,8,11,9,7,10,12);(2,4)(13,14,15,19,16,20,21,26,17,18,22,24,23,25,27,28);(5,7,8,12,9,6,10,11)(14,18)(19,24)(20,25)(26,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,e^2=g,f^2=h,h^2=i,b^a=bd,c^b=ci224 conjugacy classes — view the full character table.
| maximal subgroup of | #72286315, #72286585, #72286666, … |
| maximal normal subgroup of | #72286315, #72286585, #72286666, … |
| maximal quotient of | #72286359 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35436);