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 | #59095 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5055 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)(3,7)(4,5)(6,8)(10,14)(15,16)(17,18,22,27)(19,23,29,24)(20,30,31,25)(21,26,32,28);(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,32)(26,31);(2,5)(6,7)(17,19,22,29)(18,24,27,23)(20,26,31,28)(21,25,32,30);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,15,14,16)(17,19)(18,24)(20,28)(21,25)(22,29)(23,27)(26,31)(30,32);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,16,14,15)(17,20)(18,25)(19,28)(21,24)(22,31)(23,32)(26,29)(27,30);(17,21)(18,26)(19,25)(20,23)(22,32)(24,31)(27,28)(29,30)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7881061);