Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
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×C22⋊C16 |
| prime radical | C22×D4 | C16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
288 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,11,16,12,17,19,28,13,18,20,29,22,30,31,38,14,15,21,25,23,26,32,35,24,27,33,36,34,37,39,40);(2,5)(3,8)(6,7);(10,15)(16,25)(17,26)(18,27)(28,35)(29,36)(30,37)(38,40)Polycyclic presentation:
2^9|a^2=e,d^2=f,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,d^a=df,d^b=df152 conjugacy classes — view the full character table.
| maximal subgroup of | #72227023, #72227085, #72227106, … |
| maximal normal subgroup of | #72227023, #72227085, #72227106, … |
| maximal quotient of | #72226992 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57240);