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 |
| prime radical | #2537 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4279 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)(9,10,15,17)(11,19,13,16)(12,18,22,14)(20,21,23,24)(25,26,31,34)(27,28,29,32)(30,33,38,40)(35,36,37,39);(1,2,4,5)(3,6,7,8)(9,11,15,13)(10,16,17,19)(12,20,22,23)(14,21,18,24)(25,27,31,29)(26,32,34,28)(30,37,38,35)(33,36,40,39);(10,17)(12,22)(16,19)(20,23)(25,28,31,32)(26,27,34,29)(30,36,38,39)(33,35,40,37);(9,12)(10,18)(11,20)(13,23)(14,17)(15,22)(16,24)(19,21)(25,28)(26,27)(29,34)(30,36)(31,32)(33,35)(37,40)(38,39);(9,13,15,11)(10,19,17,16)(12,23,22,20)(14,24,18,21)(25,29,31,27)(26,28,34,32)(30,35,38,37)(33,39,40,36);(1,3,4,7)(2,6,5,8)(9,14,15,18)(10,12,17,22)(11,21,13,24)(16,20,19,23)(25,30,31,38)(26,33,34,40)(27,35,29,37)(28,36,32,39)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,e^2=h,f^2=g,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, 7879507);