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 | #2416 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(11,19)(13,21)(16,20)(17,22)(25,26,30,33)(27,31,34,38)(28,29,32,37)(35,36,39,40);(1,2,5,6)(3,7,4,8)(9,10,15,12)(11,20,19,16)(13,17,21,22)(14,18,24,23)(27,34)(31,38)(35,39)(36,40);(1,3,5,4)(2,7,6,8)(9,11,15,19)(10,16,12,20)(13,14,21,24)(17,23,22,18)(25,27)(26,31)(28,35)(29,36)(30,34)(32,39)(33,38)(37,40);(9,12,15,10)(11,16,19,20)(13,22,21,17)(14,23,24,18);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,14)(12,22)(15,21)(16,23)(18,20)(19,24)(25,28)(26,29)(27,35)(30,32)(31,36)(33,37)(34,39)(38,40);(9,14)(10,18)(11,21)(12,23)(13,19)(15,24)(16,17)(20,22)(25,29)(26,32)(27,36)(28,33)(30,37)(31,39)(34,40)(35,38)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7759947);