Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #35728 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(6,8)(9,10,13,15)(11,12,16,14)(17,18)(19,21)(20,28)(22,23)(24,26)(25,31)(27,32)(29,30);(3,7)(6,8)(9,11,13,16)(10,14,15,12)(18,23)(19,26)(25,28)(27,29);(1,2,5,4)(3,6,7,8)(9,11)(10,12)(13,16)(14,15)(18,23)(21,24)(25,28)(30,32);(1,3,5,7)(2,6,4,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,24)(20,29)(21,23)(22,26)(25,30)(27,31)(28,32);(1,4,5,2)(3,8,7,6)(17,20,22,31)(18,25,23,28)(19,27,26,29)(21,30,24,32);(17,21,22,24)(18,19,23,26)(20,30,31,32)(25,27,28,29)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=ghi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8247057);