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 |
| prime radical | #46090 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,15)(11,12,17,16)(13,18,21,22)(19,20,24,23)(25,26,31,33)(27,34,29,32)(28,37,39,30)(35,36,38,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(18,22)(19,24)(25,27,31,29)(26,32,33,34)(28,35,39,38)(30,36,37,40);(2,5)(6,7)(13,21)(18,22)(19,24)(20,23)(26,33)(30,37)(32,34)(36,40);(9,11,14,17)(10,16,15,12)(13,19,21,24)(18,23,22,20)(25,28)(26,30)(27,35)(29,38)(31,39)(32,36)(33,37)(34,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,29,31,27)(26,34,33,32)(28,38,39,35)(30,40,37,36);(1,2)(3,6)(4,5)(7,8)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,30)(26,28)(27,36)(29,40)(31,37)(32,35)(33,39)(34,38)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=ghi,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, 8095302);