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 | |
4910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,8)(3,7,4,5)(9,10,12,14)(11,13,15,16)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32);(1,3,6,4)(2,7,8,5)(9,10)(11,13)(12,14)(15,16)(19,25)(21,29)(23,26)(24,28);(2,8)(3,4)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(17,18,22,20)(19,26,25,23)(21,24,29,28)(27,32,31,30);(1,4,6,3)(2,7,8,5)(17,20,22,18)(19,23,25,26)(21,28,29,24)(27,30,31,32);(1,5,6,7)(2,3,8,4)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7646568);