Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | D8 |
| commutator | C8 | C4×C16 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1370 |
| prime radical | C23 | #364 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C22×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
162 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 86 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 — 2 generators on 36 points:
(1,2)(3,4)(5,6,9,17,12,18,25,35,13,19,29,36,28,16,26,34,14,20,22,15,8,27,24,32,11,31,23,33,10,30,7,21);(2,4)(5,7,10,23,11,24,8,22,14,26,28,29,13,25,12,9)(6,15,18,32,19,33,16,21,20,17,27,35,31,36,30,34)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=f,f^2=i,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efi,f^a=fi176 conjugacy classes — view the full character table.
| maximal subgroup of | #72216247, #72216565, #72225517, … |
| maximal normal subgroup of | #72216247, #72216565, #72225517, … |
| maximal quotient of | #72227190 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1214);