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 | |
4599 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,13)(12,15)(14,17)(16,18)(19,21)(20,23)(22,24)(25,26)(27,28)(29,37)(30,34)(31,32)(33,39)(35,40)(36,38);(1,2,5,6)(3,7,4,8)(9,11)(10,13)(12,18)(14,19)(15,16)(17,21)(20,22)(23,24)(25,27,31,30)(26,28,32,34)(29,35,39,38)(33,36,37,40);(10,12)(11,16)(17,20)(19,24)(26,32)(28,34)(33,37)(36,40);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,18,16,13)(14,20,23,17)(19,22,24,21)(25,28,31,34)(26,27,32,30)(29,36,39,40)(33,35,37,38);(9,13,15,18)(10,16,12,11)(14,22,23,21)(17,19,20,24)(25,29,31,39)(26,33,32,37)(27,35,30,38)(28,36,34,40);(1,4,5,3)(2,8,6,7)(9,14,15,23)(10,17,12,20)(11,19,16,24)(13,21,18,22)(25,30,31,27)(26,34,32,28)(29,38,39,35)(33,40,37,36)Polycyclic presentation:
2^9|b^2=ghi,d^2=gh,e^2=h,f^2=gh,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, 7859691);