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 | #51937 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,16)(11,18,13,15)(12,20,22,17)(19,23,21,24);(1,3,5,8)(2,6,7,4)(9,11,14,13)(10,15,16,18)(12,21,22,19)(17,24,20,23)(25,26,30,33)(27,34,37,31)(28,29,32,38)(35,40,39,36);(1,3)(2,4)(5,8)(6,7)(10,16)(15,18)(17,20)(23,24)(25,27,30,37)(26,31,33,34)(28,35,32,39)(29,36,38,40);(1,4)(2,3)(5,6)(7,8)(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,20)(18,24)(25,28)(26,29)(27,35)(30,32)(31,36)(33,38)(34,40)(37,39);(9,10)(11,18)(12,17)(13,15)(14,16)(19,24)(20,22)(21,23)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(9,13,14,11)(10,15,16,18)(12,19,22,21)(17,24,20,23)(25,29)(26,32)(27,36)(28,33)(30,38)(31,39)(34,35)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8514620);