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 | |
4535 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)(11,17)(12,13)(14,18)(15,21)(16,22)(19,24)(20,23)(25,26,29,33)(27,30,35,31)(28,32,37,40)(34,38,36,39);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,24)(17,18)(22,23)(25,27)(26,30)(28,36)(29,35)(31,33)(32,39)(34,37)(38,40);(1,3)(2,5)(4,8)(6,7)(12,22)(13,16)(19,23)(20,24)(25,27,29,35)(26,31,33,30)(28,34,37,36)(32,39,40,38);(3,8)(5,6)(9,12)(10,16)(11,19)(13,18)(14,22)(15,24)(17,20)(21,23)(25,27)(26,30)(28,36)(29,35)(31,33)(32,39)(34,37)(38,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,17)(12,23)(13,20)(14,21)(15,18)(16,24)(19,22)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,40)(35,36);(1,2)(3,6)(4,7)(5,8)(9,13,14,16)(10,12,18,22)(11,20,21,24)(15,19,17,23)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=h,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8809976);