Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C2×D4 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C2×C4⋊C16 |
| prime radical | C23 | C4⋊C16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
166 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,19,24,20,25,27,36,21,26,28,37,30,38,39,46,22,23,29,33,31,34,40,43,32,35,41,44,42,45,47,48);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(18,23)(24,33)(25,34)(26,35)(36,43)(37,44)(38,45)(46,48)Polycyclic presentation:
2^9|a^2=e,b^2=d,c^2=f,d^2=f,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,d^a=df176 conjugacy classes — view the full character table.
| maximal subgroup of | #72225400, #72225407, #72225409, … |
| maximal normal subgroup of | #72225400, #72225407, #72225409, … |
| maximal quotient of | #72225395 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58238);