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 | |
4426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,8)(6,7)(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,23)(21,22)(25,26,30,28)(27,31,35,33)(29,36,37,32)(34,39,40,38);(1,3)(2,6)(4,8)(5,7)(10,15)(11,18)(16,23)(19,21)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,2)(3,6)(4,7)(5,8)(10,15)(12,20)(13,17)(19,21)(26,28)(29,37)(31,33)(34,40);(1,4,5,6)(2,7,8,3)(9,11)(10,13)(12,21)(14,18)(15,17)(16,24)(19,20)(22,23)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,4)(2,3)(5,6)(7,8)(9,12,14,20)(10,16,15,23)(11,19,18,21)(13,22,17,24)(25,28,30,26)(27,31,35,33)(29,36,37,32)(34,38,40,39);(1,3)(2,4)(5,7)(6,8)(9,13)(10,11)(12,22)(14,17)(15,18)(16,19)(20,24)(21,23)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948190);