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 | |
4714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,19,12)(13,16,21,22)(18,23,24,20)(27,28)(31,32)(35,36)(38,39);(1,2)(3,7)(4,5)(6,8)(10,15)(12,17)(16,22)(20,23)(25,26,30,34)(27,32,28,31)(29,33,37,40)(35,39,36,38);(2,5)(6,7)(13,21)(16,22)(18,24)(20,23)(25,27,30,28)(26,31,34,32)(29,36,37,35)(33,39,40,38);(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3,4,8)(2,6,5,7)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,28,30,27)(26,32,34,31)(29,36,37,35)(33,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,e^2=i,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, 8121250);