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 | #37832 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5301 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,10,12,14)(11,13,15,16)(17,18,22,25)(19,23,28,31)(20,21,24,29)(26,27,30,32);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,19,22,28)(18,23,25,31)(20,26,24,30)(21,27,29,32);(1,4)(2,3)(5,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32);(1,3)(2,6)(4,8)(5,7);(1,3)(2,4)(5,7)(6,8)(17,20)(18,21)(19,26)(22,24)(23,27)(25,29)(28,30)(31,32);(1,4)(2,7)(3,8)(5,6)(17,21)(18,24)(19,27)(20,25)(22,29)(23,30)(26,31)(28,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7764305);