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 | #2430 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(10,13)(14,16)(17,18,23,26)(19,27,29,24)(20,30,28,22)(21,25,31,32);(1,2)(3,8)(4,6)(5,7)(11,15)(14,16)(17,19,23,29)(18,24,26,27)(20,21,28,31)(22,25,30,32);(1,3,5,6)(2,4,7,8)(9,10)(11,14)(12,13)(15,16);(1,2)(3,4)(5,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,20,23,28)(18,22,26,30)(19,21,29,31)(24,25,27,32);(1,2)(3,4)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,21)(18,25)(19,28)(20,29)(22,27)(23,31)(24,30)(26,32);(1,4)(2,6)(3,7)(5,8)(17,22)(18,20)(19,25)(21,24)(23,30)(26,28)(27,31)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8571656);