Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×M16 | C24 |
| socle | C2 | #17363 |
| prime radical | #3444 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C4×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
707 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,10,5,11,16,25,6,7,15,19,17,8,20,22)(3,9,12,23,13,24,27,32,14,18,26,29,28,21,30,31);(2,7)(8,11)(9,18)(21,24);(2,8,7,11)(4,15)(9,21,18,24)(10,25,19,22)(12,26)(16,20)(23,32,29,31)(27,30);(1,3,5,13,6,14,17,28)(2,9,11,24,7,18,8,21)(4,12,16,27,15,26,20,30)(10,23,25,32,19,29,22,31)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=h,f^2=h,h^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,f^b=fg,f^c=fi,f^e=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287622, #72287633, #72287815, … |
| maximal normal subgroup of | #72287622, #72287633, #72287815, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387632);