Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | D16 |
| commutator | C16 | C2×C16 |
| Frattini, cosocle | C8×C16 | C22 |
| socle | C2 | #7100 |
| prime radical | C22 | #1373 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×C16 ◁ C4×C16 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
59 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,13,7,15,4,27,9,16,29,44,28,14,31,45,10,17,33,46,6,34,32,62,8,36,19,12,35,42,30,43)(3,18,21,48,23,50,20,11,25,37,54,39,53,49,56,40,26,51,58,38,22,59,57,41,24,61,52,47,60,64,55,63);(1,3,19,52,6,22,31,56,9,25,5,21,35,60,32,57,10,26,29,54,7,23,30,55,8,24,33,58,28,53,4,20)(2,11,27,49,14,38,46,61,36,63,43,50,15,39,44,51,17,41,62,64,42,48,13,37,16,40,45,59,34,47,12,18)Polycyclic presentation:
2^9|a^2=d,b^2=ce,c^2=eg,d^2=f,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi152 conjugacy classes — view the full character table.
| maximal subgroup of | #72225352, #72225355, #72225356, … |
| maximal normal subgroup of | #72225352, #72225355, #72225356, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1078);