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 | |
4644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,15)(13,21)(14,17)(16,22)(19,23)(20,24)(25,26)(27,33)(28,29)(30,34)(31,32)(35,40)(36,37)(38,39);(1,2)(3,7)(4,5)(6,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(26,32)(28,33)(34,39)(36,40);(2,5)(6,7)(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24)(25,27,31,29)(26,33,32,28)(30,37,38,35)(34,36,39,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24)(25,28)(26,27)(29,32)(30,36)(31,33)(34,35)(37,39)(38,40);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,29,31,27)(26,28,32,33)(30,37,38,35)(34,36,39,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)Polycyclic presentation:
2^9|c^2=gh,d^2=h,e^2=g,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, 8083859);