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 | |
4541 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,17)(11,12,19,15)(13,16,21,23)(18,20,24,22);(3,7)(6,8)(9,11,14,19)(10,15,17,12)(13,18,21,24)(16,22,23,20)(25,26)(27,32)(28,36)(29,34)(30,31)(33,38)(35,37)(39,40);(1,2,5,4)(3,6,7,8)(13,21)(16,23)(18,24)(20,22)(26,31)(29,32)(33,36)(37,39);(1,3,5,7)(2,6,4,8)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,22)(23,24)(25,27,30,34)(26,32,31,29)(28,35,38,40)(33,39,36,37);(1,4,5,2)(3,8,7,6)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,28)(26,33)(27,35)(29,37)(30,38)(31,36)(32,39)(34,40);(9,13,14,21)(10,16,17,23)(11,18,19,24)(12,20,15,22)(25,29,30,32)(26,34,31,27)(28,37,38,39)(33,40,36,35)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,d^2=hi,e^2=h,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=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8149508);