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 | |
4509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,17)(12,20)(13,18)(14,15)(16,22)(19,21)(23,24)(25,26,31,33)(27,32,35,40)(28,30,38,39)(29,37,36,34);(2,6)(3,8)(10,15)(13,17)(16,20)(21,23)(25,27,31,35)(26,32,33,40)(28,36,38,29)(30,34,39,37);(1,3)(2,4)(5,8)(6,7)(10,15)(11,18)(16,20)(19,24)(26,33)(27,35)(28,38)(34,37);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,23)(22,24)(25,28)(26,30)(27,36)(29,35)(31,38)(32,34)(33,39)(37,40);(1,2)(3,4)(5,6)(7,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,20)(17,23)(18,24)(25,29,31,36)(26,34,33,37)(27,28,35,38)(30,32,39,40);(1,4)(2,3)(5,7)(6,8)(9,13)(10,11)(12,21)(14,17)(15,18)(16,19)(20,24)(22,23)(25,30)(26,28)(27,37)(29,32)(31,39)(33,38)(34,35)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8659697);