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 | #59026 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,4)(5,7)(6,8)(10,12)(14,16)(17,18,23,26)(19,27,30,24)(20,31,29,22)(21,32,28,25);(3,8)(4,6)(11,15)(14,16)(17,19)(18,24)(20,28)(21,29)(22,25)(23,30)(26,27)(31,32);(1,3,5,8)(2,6,7,4)(9,10,13,12)(11,14,15,16);(1,4)(2,3)(5,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20,23,29)(18,22,26,31)(19,28,30,21)(24,25,27,32);(9,12,13,10)(11,16,15,14)(17,21,23,28)(18,25,26,32)(19,29,30,20)(22,27,31,24);(1,2)(3,6)(4,8)(5,7)(17,22)(18,20)(19,25)(21,27)(23,31)(24,28)(26,29)(30,32)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8345750);