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 | C2×#532 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,6)(4,8)(5,7)(9,10,12,14)(11,13,15,16)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,31,32,30);(1,3,5,4)(2,6,7,8)(9,10)(11,13)(12,14)(15,16)(18,19)(23,25)(24,26)(30,31);(2,7)(6,8)(9,11,12,15)(10,13,14,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30);(1,2)(3,6)(4,8)(5,7)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(1,4,5,3)(2,8,7,6)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7581661);