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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4317 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,13,18,16)(12,20,21,17)(19,24,22,23)(25,26,30,33)(27,31,35,28)(29,32,38,40)(34,37,39,36);(1,2)(3,7)(4,5)(6,8)(10,15)(11,18)(12,21)(23,24)(25,27,30,35)(26,28,33,31)(29,34,38,39)(32,36,40,37);(2,5)(6,7)(10,15)(13,16)(17,20)(23,24)(25,28,30,31)(26,27,33,35)(29,37,38,36)(32,39,40,34);(3,8)(6,7)(9,11,14,18)(10,16,15,13)(12,22,21,19)(17,23,20,24)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,39)(38,40);(1,3,4,8)(2,6,5,7)(9,12,14,21)(10,17,15,20)(11,19,18,22)(13,23,16,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,37)(36,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,13,14,16)(10,18,15,11)(12,23,21,24)(17,22,20,19)(25,29,30,38)(26,32,33,40)(27,34,35,39)(28,36,31,37)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8269375);