Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347092 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35050 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3489 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,11)(4,15)(5,9)(6,8)(7,12)(10,18)(13,25)(14,24)(16,21)(17,28)(19,22)(20,26)(23,30)(27,32)(29,31);(1,3,5,14)(2,7,9,21)(4,16,18,12)(6,13,19,27)(8,20,22,31)(10,24,15,11)(17,29,30,26)(23,32,28,25);(4,17)(10,23)(11,25)(12,26)(15,28)(16,29)(18,30)(24,32);(2,8)(7,20)(9,22)(10,23)(11,25)(15,28)(21,31)(24,32);(2,9)(7,21)(8,22)(10,15)(11,24)(20,31)(23,28)(25,32)Polycyclic presentation:
2^9|b^2=h,f^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,f^b=fh,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284841, #72284955, #72285734, … |
| maximal normal subgroup of | #72284841, #72284955, #72285734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254289);