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 | |
4282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(7,8)(9,10,14,17)(11,18,15,12)(13,21,22,16)(19,20,23,24)(25,26)(27,35)(28,33)(29,34)(30,31)(32,38)(36,40)(37,39);(1,2,6,5)(3,7,4,8)(9,11,14,15)(10,12,17,18)(13,19,22,23)(16,20,21,24)(26,31)(32,35)(33,34)(39,40);(1,3,6,4)(2,7,5,8)(9,11,14,15)(10,12,17,18)(13,19,22,23)(16,20,21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(1,2,6,5)(3,8,4,7)(9,12,14,18)(10,15,17,11)(13,20,22,24)(16,23,21,19)(25,27,30,38)(26,32,31,35)(28,36,29,37)(33,39,34,40);(1,4,6,3)(2,8,5,7)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,22)(23,24)(25,28,30,29)(26,33,31,34)(27,36,38,37)(32,39,35,40);(1,5,6,2)(3,8,4,7)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,29,30,28)(26,34,31,33)(27,37,38,36)(32,40,35,39)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066303);