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 | #59755 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4384 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)(3,7)(4,6)(5,8)(9,10,14,15)(11,13,18,16)(12,20,22,17)(19,23,24,21)(27,33)(28,36)(31,34)(32,37);(1,2,4,6)(3,5,8,7)(10,15)(11,18)(17,20)(19,24)(25,26,30,29)(27,34,33,31)(28,32,36,37)(35,40,39,38);(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,39)(37,40);(1,3,4,8)(2,5,6,7)(9,11,14,18)(10,16,15,13)(12,19,22,24)(17,23,20,21);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,28,30,36)(26,32,29,37)(27,35,33,39)(31,38,34,40);(9,13,14,16)(10,18,15,11)(12,21,22,23)(17,24,20,19)(25,29,30,26)(27,31,33,34)(28,37,36,32)(35,38,39,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8087660);