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 | |
4493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,16)(12,21)(13,14)(15,17)(18,24)(19,23)(20,22)(27,34)(32,38)(35,40)(36,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,17,21)(13,19,24,22)(14,23,18,20)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,39)(36,40);(1,3,4,7)(2,6,5,8)(10,17)(11,12)(13,24)(20,23)(26,31)(28,37)(32,38)(35,40);(9,12,15,11)(10,16,17,21)(13,22,24,19)(14,23,18,20)(25,27)(26,32)(28,35)(29,36)(30,34)(31,38)(33,39)(37,40);(9,13,15,24)(10,18,17,14)(11,19,12,22)(16,20,21,23)(25,28,30,37)(26,33,31,29)(27,35,34,40)(32,39,38,36);(9,14,15,18)(10,13,17,24)(11,20,12,23)(16,22,21,19)(25,29,30,33)(26,28,31,37)(27,36,34,39)(32,35,38,40)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7895197);