Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #2819 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,7)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,22,25,21,23,26,30)(19,24,27,31,20,28,29,32);(2,6)(3,4)(9,10)(11,14)(12,13)(15,16)(18,23)(24,28)(25,30)(31,32);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,19,26,29,21,20,22,27)(18,24,30,32,23,28,25,31);(1,4,5,3)(2,8,6,7)(9,11,12,16)(10,14,13,15)(18,23)(19,20)(25,30)(27,29);(2,6)(7,8)(17,20,22,29,21,19,26,27)(18,24,25,31,23,28,30,32)Polycyclic presentation:
2^9|a^2=h,c^2=fh,d^2=g,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dfi,d^c=di,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338626);