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 | |
4454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(27,28)(31,32)(35,36)(38,39);(1,2,4,6)(3,5,8,7)(9,10,14,15)(11,13,17,22)(12,19,21,16)(18,24,23,20)(25,26,30,34)(27,32,28,31)(29,33,37,40)(35,39,36,38);(1,2)(3,5)(4,6)(7,8)(10,15)(13,22)(16,19)(20,24)(25,27,30,28)(26,31,34,32)(29,36,37,35)(33,39,40,38);(1,3)(2,5)(4,8)(6,7)(9,11,14,17)(10,13,15,22)(12,18,21,23)(16,20,19,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(9,12,14,21)(10,16,15,19)(11,18,17,23)(13,20,22,24)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(9,13)(10,17)(11,15)(12,20)(14,22)(16,23)(18,19)(21,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=ghi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8134601);