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 | |
4634 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)(11,15)(12,20)(13,14)(16,21)(17,19)(18,23)(22,24)(25,26,30,31)(27,28,34,35)(29,32,37,38)(33,36,39,40);(1,2,5,6)(3,4,7,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,22)(23,24)(26,31)(28,35)(29,37)(33,39);(2,6)(3,7)(10,14)(15,17)(16,20)(22,23)(25,27,30,34)(26,28,31,35)(29,33,37,39)(32,36,38,40);(2,6)(3,7)(9,11,13,19)(10,15,14,17)(12,18,21,24)(16,22,20,23)(25,28,30,35)(26,27,31,34)(29,36,37,40)(32,33,38,39);(1,3)(2,4)(5,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27)(26,28)(29,33)(30,34)(31,35)(32,36)(37,39)(38,40);(1,4)(2,3)(5,8)(6,7)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29)(26,32)(27,33)(28,36)(30,37)(31,38)(34,39)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=g,d^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, 10236076);