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 | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4901 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(9,10,12,14)(11,13,15,16)(17,18,23,27)(19,25,20,24)(21,22,26,32)(28,31,30,29);(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16);(1,3)(2,5)(4,8)(6,7)(17,19,23,20)(18,24,27,25)(21,28,26,30)(22,29,32,31);(17,20,23,19)(18,25,27,24)(21,30,26,28)(22,31,32,29);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,13,14,16)(17,21)(18,22)(19,28)(20,30)(23,26)(24,29)(25,31)(27,32);(1,2)(3,5)(4,7)(6,8)(17,22)(18,26)(19,29)(20,31)(21,27)(23,32)(24,30)(25,28)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7561022);