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 | |
4981 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,30,31)(27,33,34,32)(28,29,37,39)(35,36,38,40);(2,6)(3,7)(11,17)(15,21)(18,23)(22,24)(26,31)(28,37)(32,33)(35,38);(1,3,5,7)(2,4,6,8)(10,14)(15,21)(16,19)(22,24)(26,31)(27,34)(29,39)(35,38);(1,4)(2,7)(3,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27)(26,32)(28,38)(29,36)(30,34)(31,33)(35,37)(39,40);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,28)(26,29)(27,35)(30,37)(31,39)(32,40)(33,36)(34,38);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29)(26,28)(27,36)(30,39)(31,37)(32,38)(33,35)(34,40)Polycyclic presentation:
2^9|a^2=gi,c^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9350374);