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 | |
4630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10)(11,18)(12,13)(14,15)(16,21)(17,22)(19,24)(20,23)(25,26)(27,34)(28,36)(29,31)(30,32)(33,37)(35,39)(38,40);(2,5)(6,8)(10,15)(12,17)(16,18)(19,23)(26,29)(28,37)(32,34)(35,38);(1,2)(3,6)(4,5)(7,8)(25,27,31,30)(26,32,29,34)(28,35,37,38)(33,39,36,40);(9,11,14,21)(10,16,15,18)(12,19,17,23)(13,20,22,24)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,17)(10,13,15,22)(11,19,21,23)(16,20,18,24)(25,29,31,26)(27,34,30,32)(28,36,37,33)(35,40,38,39);(1,3)(2,6)(4,7)(5,8)(9,13)(10,17)(11,20)(12,15)(14,22)(16,23)(18,19)(21,24)(25,30,31,27)(26,34,29,32)(28,38,37,35)(33,40,36,39)Polycyclic presentation:
2^9|c^2=h,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015615);