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 | |
4492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(5,8)(13,21)(17,22)(18,23)(20,24)(25,26)(27,33)(28,37)(29,38)(30,31)(32,36)(34,35)(39,40);(2,4)(3,7)(10,15)(12,19)(17,22)(20,24)(26,31)(28,38)(32,33)(34,40);(1,2,6,4)(3,5,7,8)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(25,27,30,36)(26,32,31,33)(28,34,38,40)(29,35,37,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,16)(10,12,15,19)(13,18,21,23)(17,20,22,24)(25,28)(26,29)(27,34)(30,38)(31,37)(32,35)(33,39)(36,40);(1,4,6,2)(3,8,7,5)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,28)(26,29)(27,34)(30,38)(31,37)(32,35)(33,39)(36,40);(1,5,6,8)(2,7,4,3)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,29)(26,28)(27,35)(30,37)(31,38)(32,34)(33,40)(36,39)Polycyclic presentation:
2^9|c^2=gi,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017090);