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 | |
4562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(26,29)(27,28)(34,37)(35,36);(1,2,5,4)(3,7,8,6)(9,10,14,13)(11,17,19,15)(12,16,21,20)(18,23,24,22)(25,26,31,29)(27,33,28,32)(30,34,38,37)(35,40,36,39);(1,3,5,8)(2,6,4,7)(10,13)(15,17)(16,20)(22,23)(25,27,31,28)(26,32,29,33)(30,35,38,36)(34,39,37,40);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,28,31,27)(26,33,29,32)(30,36,38,35)(34,40,37,39);(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,29,31,26)(27,33,28,32)(30,37,38,34)(35,40,36,39);(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,23,24,22)(25,30,31,38)(26,34,29,37)(27,35,28,36)(32,39,33,40)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8104458);