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 | |
4564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10,14,15)(11,18,17,13)(12,20,21,16)(19,23,22,24)(25,26,30,33)(27,31,34,28)(29,32,38,40)(35,39,37,36);(2,5)(3,7)(10,15)(12,21)(13,18)(19,22)(25,27,30,34)(26,31,33,28)(29,37,38,35)(32,36,40,39);(1,2)(3,6)(4,5)(7,8)(10,15)(11,17)(16,20)(19,22)(27,34)(28,31)(35,37)(36,39);(2,5)(6,8)(9,11,14,17)(10,13,15,18)(12,22,21,19)(16,24,20,23)(25,28)(26,27)(29,36)(30,31)(32,35)(33,34)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,19)(13,23)(14,21)(15,20)(17,22)(18,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,37)(38,40);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,15)(12,23)(14,18)(16,22)(19,20)(21,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,40)(34,37)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8735890);