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 | |
4511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,14,15)(11,13,18,16)(12,20,21,17)(19,24,22,23)(25,26,31,35)(27,32,36,39)(28,29,33,40)(30,34,37,38);(1,2,5,4)(3,7,8,6)(10,15)(11,18)(12,21)(23,24)(25,27,31,36)(26,32,35,39)(28,37,33,30)(29,38,40,34);(1,3,5,8)(2,6,4,7)(10,15)(13,16)(17,20)(23,24)(27,36)(30,37)(32,39)(34,38);(1,4,5,2)(3,7,8,6)(9,11,14,18)(10,16,15,13)(12,22,21,19)(17,23,20,24)(25,28)(26,29)(27,37)(30,36)(31,33)(32,38)(34,39)(35,40);(9,12)(10,17)(11,19)(13,23)(14,21)(15,20)(16,24)(18,22)(25,29)(26,33)(27,34)(28,35)(30,39)(31,40)(32,37)(36,38);(9,13,14,16)(10,18,15,11)(12,23,21,24)(17,22,20,19)(25,30,31,37)(26,34,35,38)(27,28,36,33)(29,39,40,32)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8282837);