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 | #62810 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2988 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)(4,12)(5,8)(7,13)(10,14,16,15)(18,22,24,27)(23,29,30,32);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,18)(19,23)(20,27)(21,24)(22,28)(25,30)(26,32)(29,31);(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31);(19,25)(23,30)(26,31)(29,32);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)Polycyclic presentation:
2^9|a^2=g,e^2=h,f^2=i,b^a=bf,e^a=eh,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, 6250685);