Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | D8 |
| commutator | C8 | C4×C16 |
| Frattini, cosocle | C2×C82 | C22 |
| socle, prime radical | C23 | #335 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
182 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 512 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2)(3,4)(6,11)(7,15)(8,17)(9,16)(12,13)(14,20)(18,19)(21,22,23,26,24,27,29,32,25,28,30,33,31,34,35,36);(2,4)(5,6,8,12,9,13,7,11,10,14,15,19,16,18,17,20)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=f,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efh,f^a=fh176 conjugacy classes — view the full character table.
| maximal subgroup of | #72225375, #72225384, #72225386, … |
| maximal normal subgroup of | #72225375, #72225384, #72225386, … |
| maximal quotient of | #72225373 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107);