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 | |
4130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,15)(13,16)(14,20)(17,23)(19,21)(22,24)(25,26,31,35)(27,29,33,40)(28,34,30,32)(36,39,37,38);(1,2)(3,8)(4,5)(6,7)(9,11,15,16)(10,13,12,18)(14,22,23,19)(17,24,20,21)(25,27)(26,29)(28,36)(30,37)(31,33)(32,38)(34,39)(35,40);(2,7)(3,4)(10,12)(13,18)(17,20)(21,24)(25,28,31,30)(26,32,35,34)(27,36,33,37)(29,38,40,39);(1,3,6,4)(2,8,7,5)(9,12,15,10)(11,18,16,13)(14,20,23,17)(19,24,22,21);(1,4,6,3)(2,5,7,8)(9,13,15,18)(10,16,12,11)(14,21,23,24)(17,22,20,19)(25,29)(26,33)(27,35)(28,38)(30,39)(31,40)(32,37)(34,36);(1,5)(2,3)(4,7)(6,8)(9,14,15,23)(10,17,12,20)(11,19,16,22)(13,21,18,24)(25,30,31,28)(26,34,35,32)(27,37,33,36)(29,39,40,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8706364);