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 | |
4498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,18,16)(12,13,21,22)(19,23,24,20)(26,31)(28,35)(32,37)(36,39);(1,2)(3,7)(4,5)(6,8)(10,15)(11,18)(13,22)(19,24)(25,26,30,31)(27,28,34,35)(29,37,38,32)(33,39,40,36);(2,5)(6,7)(10,15)(12,21)(16,17)(19,24)(25,27,30,34)(26,28,31,35)(29,33,38,40)(32,36,37,39);(2,5)(6,7)(9,11,14,18)(10,16,15,17)(12,19,21,24)(13,20,22,23)(25,28,30,35)(26,27,31,34)(29,36,38,39)(32,33,37,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,13)(11,19)(14,21)(15,22)(16,20)(17,23)(18,24)(25,27)(26,28)(29,33)(30,34)(31,35)(32,36)(37,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,12)(11,20)(14,22)(15,21)(16,19)(17,24)(18,23)(25,29,30,38)(26,32,31,37)(27,33,34,40)(28,36,35,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10230140);