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 | |
4328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(25,26)(27,33)(28,30)(29,36)(31,34)(32,37)(35,38)(39,40);(1,3,6,5)(2,7,4,8)(9,10)(11,16)(12,13)(14,15)(17,22)(18,24)(19,21)(20,23)(26,28)(27,34)(32,36)(35,40);(1,2,6,4)(3,7,5,8)(10,15)(13,17)(16,23)(19,24)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,3,6,5)(2,7,4,8)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,24)(18,22)(25,28,30,26)(27,33,34,31)(29,36,37,32)(35,39,40,38);(1,4,6,2)(3,7,5,8)(9,12)(10,17)(11,18)(13,15)(14,22)(16,19)(20,21)(23,24)(25,29,30,37)(26,32,28,36)(27,35,34,40)(31,38,33,39);(1,5,6,3)(2,7,4,8)(9,13,14,17)(10,12,15,22)(11,19,20,24)(16,21,23,18)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=g,e^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, 8278764);