Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343485 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(18,22)(20,27)(23,29)(24,28)(26,31)(30,32);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,18,21,24)(19,23,25,30)(20,28,27,22)(26,32,31,29);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,21,25)(18,23,24,30)(20,26,27,31)(22,29,28,32);(19,25)(23,30)(26,31)(29,32);(2,8)(6,9)(7,14)(13,15)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,f^2=i,b^a=bf,e^a=ehi,f^a=fh,f^b=fi,g^b=ghi,d^c=di98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250825);