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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,4)(3,8,5,7)(17,18)(19,27)(20,26)(21,24)(22,28)(23,25)(29,30)(31,32);(1,3,6,5)(2,7,4,8)(9,10)(11,14)(12,15)(13,16)(17,19,23,21)(18,24,25,27)(20,29,28,31)(22,30,26,32);(1,2,6,4)(3,8,5,7)(9,11)(10,12)(13,15)(14,16)(18,25)(20,28)(24,27)(29,31);(17,20,23,28)(18,26,25,22)(19,29,21,31)(24,32,27,30);(1,4,6,2)(3,8,5,7)(9,11)(10,14)(12,16)(13,15)(17,21,23,19)(18,27,25,24)(20,31,28,29)(22,32,26,30);(1,5,6,3)(2,7,4,8)(9,12,13,14)(10,15,16,11)(17,22,23,26)(18,28,25,20)(19,30,21,32)(24,31,27,29)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=hi,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, 7786453);