Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C25×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C42 | C23 |
| commutator | C22 | C2×C82 |
| socle | C24 | C2×C42 |
| prime radical | C25 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1264 conjugacy classes of subgroups; only the 434 normal subgroups are shown.
This group has 1264 conjugacy classes of subgroups — too many to draw. Only its 434 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(6,8)(9,10,12,15,13,11,14,16);(2,4)(17,18,19,21,20,22,23,24);(5,6,7,8)(9,11,12,16,13,10,14,15)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,f^2=h,g^2=i,b^a=bd,c^a=ce224 conjugacy classes — view the full character table.
| maximal subgroup of | #72214267, #72214375, #72214432, … |
| maximal normal subgroup of | #72214267, #72214375, #72214432, … |
| maximal quotient of | #72224867 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17238);