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 | #49578 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10)(11,20)(12,13)(14,19)(15,17)(16,22)(18,23)(21,24)(25,26)(27,31)(28,37)(29,34)(30,35)(32,38)(33,39)(36,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,22)(10,16,17,20)(12,19,18,24)(13,21,23,14)(25,27)(26,31)(28,36)(29,33)(30,38)(32,35)(34,39)(37,40);(10,17)(12,18)(16,20)(19,24)(25,27,30,38)(26,32,35,31)(28,34,39,36)(29,37,40,33);(1,3,5,4)(2,7,6,8)(9,12,15,18)(10,13,17,23)(11,19,22,24)(14,16,21,20)(25,27)(26,32)(28,36)(29,37)(30,38)(31,35)(33,40)(34,39);(9,13)(10,18)(11,21)(12,17)(14,22)(15,23)(16,24)(19,20)(25,28)(26,33)(27,36)(29,31)(30,39)(32,40)(34,38)(35,37);(1,4,5,3)(2,8,6,7)(9,14,15,21)(10,19,17,24)(11,13,22,23)(12,16,18,20)(25,29,30,40)(26,34,35,36)(27,37,38,33)(28,32,39,31)Polycyclic presentation:
2^9|b^2=hi,c^2=i,d^2=g,f^2=gi,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, 7893198);