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 |
| prime radical | #59142 | C2 |
| 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:
(10,15)(11,17)(16,19)(18,23)(25,26)(27,32)(28,29)(30,35)(31,33)(34,39)(36,40)(37,38);(2,5)(6,8)(9,10,14,15)(11,13,17,22)(12,19,20,16)(18,24,23,21)(25,27,31,36)(26,32,33,40)(28,37,34,35)(29,38,39,30);(1,2,4,5)(3,6,7,8)(12,20)(16,19)(18,23)(21,24)(26,33)(27,36)(29,39)(35,37);(3,7)(6,8)(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,24)(20,23)(25,28,31,34)(26,29,33,39)(27,35,36,37)(30,40,38,32);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,19)(17,23)(22,24)(25,29)(26,34)(27,30)(28,33)(31,39)(32,37)(35,40)(36,38);(1,3)(2,6)(4,7)(5,8)(9,13)(10,17)(11,15)(12,21)(14,22)(16,23)(18,19)(20,24)(25,30)(26,35)(27,29)(28,32)(31,38)(33,37)(34,40)(36,39)Polycyclic presentation:
2^9|b^2=gh,c^2=hi,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8506513);