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 | |
4730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,7)(3,4,8,6)(10,13)(14,16)(17,18,23,25)(19,22,28,24)(20,30,31,26)(21,32,29,27);(9,10)(11,14)(12,13)(15,16)(17,19,23,28)(18,24,25,22)(20,29,31,21)(26,27,30,32);(1,3,5,8)(2,6,7,4)(18,25)(19,28)(20,31)(27,32);(1,3)(2,4)(5,8)(6,7)(17,20)(18,26)(19,29)(21,28)(22,32)(23,31)(24,27)(25,30);(1,4)(2,3)(5,6)(7,8)(9,11,12,15)(10,14,13,16)(17,21,23,29)(18,27,25,32)(19,20,28,31)(22,26,24,30);(17,22,23,24)(18,28,25,19)(20,32,31,27)(21,30,29,26)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7847490);