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:
(1,2,5,3)(4,7,8,6)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26,30,31)(27,35,36,32)(28,33,39,38)(29,37,40,34);(2,3)(6,7)(12,19)(16,22)(18,23)(20,24)(26,31)(28,39)(32,35)(34,37);(1,3,5,2)(4,7,8,6)(10,15)(11,17)(16,22)(18,23)(26,31)(27,36)(28,39)(29,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,32)(28,34)(29,38)(30,36)(31,35)(33,40)(37,39);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,28)(26,33)(27,37)(29,32)(30,39)(31,38)(34,36)(35,40);(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,29)(26,34)(27,38)(28,32)(30,40)(31,37)(33,36)(35,39)Polycyclic presentation:
2^9|a^2=hi,c^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,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, 9399542);