Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C8×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
363 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(6,10,12,15)(7,13,14,16)(21,22,24,28,25,29,32,35,26,23,27,30,33,31,34,36);(1,2,3,4)(5,6,8,15,9,12,17,10)(7,11,16,19,14,20,13,18)(21,23,24,30,25,31,32,36,26,22,27,28,33,29,34,35);(5,7,9,14)(6,11,12,20)(8,16,17,13)(10,18,15,19)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=g,d^2=g,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=eg,f^c=fg152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43853);