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 | #38826 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,13)(14,19)(15,17)(18,24)(20,21)(22,23)(25,26)(27,35)(28,39)(29,33)(30,34)(31,38)(32,37)(36,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,20)(13,21)(14,17)(15,19)(18,23)(22,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,40)(33,39)(34,35);(10,17)(11,19)(12,18)(21,22)(25,26,30,34)(27,31,38,35)(28,32,37,39)(29,33,36,40);(1,2,5,4)(3,6,8,7)(9,12,15,18)(10,13,17,24)(11,20,19,23)(14,22,16,21)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,40)(37,39);(1,2,5,4)(3,6,8,7)(9,13)(10,18)(11,21)(12,17)(14,20)(15,24)(16,23)(19,22)(25,28,30,37)(26,32,34,39)(27,36,38,29)(31,40,35,33);(1,4,5,2)(3,7,8,6)(9,14)(10,19)(11,17)(12,22)(13,23)(15,16)(18,21)(20,24)(25,29,30,36)(26,33,34,40)(27,37,38,28)(31,39,35,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=gh,e^2=g,f^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, 7773660);