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 | |
4643 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)(4,5)(6,8)(9,10)(11,17)(12,13)(14,15)(16,18)(19,20)(21,23)(22,24)(26,30)(28,36)(31,33)(34,40);(2,5)(6,7)(10,15)(11,18)(12,21)(20,24)(25,26,29,30)(27,31,35,33)(28,32,36,37)(34,38,40,39);(1,2)(3,6)(4,5)(7,8)(11,18)(12,21)(13,23)(16,17)(25,26,29,30)(27,33,35,31)(28,37,36,32)(34,38,40,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,22)(13,20)(14,18)(15,17)(19,21)(23,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(9,12)(10,13)(11,19)(14,21)(15,23)(16,24)(17,20)(18,22)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(9,13)(10,12)(11,20)(14,23)(15,21)(16,22)(17,19)(18,24)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|b^2=h,c^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965456);