Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | #2714 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1467 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,25,22,27,20,30)(19,21,28,32,29,24,26,31);(1,3,6,2)(4,7,5,8)(9,11,12,15)(10,13,14,16)(17,19,22,29)(18,24,27,21)(20,26,23,28)(25,31,30,32);(10,14)(13,16)(17,19)(18,21)(20,26)(22,29)(23,28)(24,27)(25,32)(30,31);(1,4,6,5)(2,7,3,8)(17,20,22,23)(18,25,27,30)(19,28,29,26)(21,31,24,32);(1,5,6,4)(2,7,3,8)(17,21,20,31,22,24,23,32)(18,26,30,29,27,28,25,19)Polycyclic presentation:
2^9|a^2=h,b^2=fg,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ci,d^b=df,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6310114);