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 | #40486 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,5)(3,6,8,7)(10,13)(14,16)(18,23)(20,29)(24,25)(26,32);(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,25,28,24)(20,21,29,31)(26,30,32,27);(2,5)(6,7)(9,11,12,15)(10,14,13,16)(17,19,22,28)(18,24,23,25)(20,26,29,32)(21,27,31,30);(1,2)(3,6)(4,5)(7,8)(17,20)(18,21)(19,26)(22,29)(23,31)(24,27)(25,30)(28,32);(1,2)(3,7)(4,5)(6,8)(17,19)(18,25)(20,26)(21,30)(22,28)(23,24)(27,31)(29,32);(1,3)(2,6)(4,8)(5,7)(17,21)(18,20)(19,27)(22,31)(23,29)(24,26)(25,32)(28,30)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=gh,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, 7876890);