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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4173 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,13)(12,20)(14,15)(16,18)(17,23)(19,24)(21,22)(27,29)(31,33)(35,37)(39,40);(2,5)(6,7)(10,15)(11,18)(17,20)(19,21)(25,26,30,34)(27,31,29,33)(28,36,38,32)(35,40,37,39);(1,2)(3,6)(4,5)(7,8)(10,15)(13,16)(17,20)(22,24)(25,27,30,29)(26,31,34,33)(28,35,38,37)(32,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,15)(14,18)(17,22)(19,23)(20,24)(25,28,30,38)(26,32,34,36)(27,35,29,37)(31,39,33,40);(1,3,4,8)(2,6,5,7)(9,12,14,23)(10,17,15,20)(11,19,18,21)(13,22,16,24)(25,29,30,27)(26,33,34,31)(28,37,38,35)(32,40,36,39);(9,13,14,16)(10,11,15,18)(12,22,23,24)(17,19,20,21)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8231301);