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 | #45563 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10,14,16)(11,15,18,12)(13,17,22,23)(19,21,24,20)(25,26,31,33)(27,29,36,32)(28,37,40,34)(30,35,38,39);(3,7)(6,8)(9,11)(10,15)(12,16)(13,19)(14,18)(17,21)(20,23)(22,24)(25,27,31,36)(26,32,33,29)(28,38,40,30)(34,39,37,35);(1,2)(3,6)(4,5)(7,8)(10,16)(11,18)(17,23)(19,24)(26,33)(29,32)(34,37)(35,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,21)(14,15)(16,18)(17,24)(19,23)(20,22)(25,28,31,40)(26,34,33,37)(27,30,36,38)(29,39,32,35);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,17)(14,16)(15,18)(19,20)(21,24)(22,23)(25,29,31,32)(26,27,33,36)(28,39,40,35)(30,37,38,34);(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,30)(26,35)(27,28)(29,37)(31,38)(32,34)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,d^2=g,e^2=g,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, 8494394);