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 | #45701 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4456 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)(11,18)(12,16)(13,22)(14,20)(15,17)(19,23)(21,24)(25,26)(27,32)(28,30)(29,33)(31,35)(34,38)(36,40)(37,39);(1,2)(3,4)(5,6)(7,8)(9,11,15,12)(10,16,17,18)(13,21,23,20)(14,19,24,22)(25,27,31,38)(26,32,35,34)(28,39,29,36)(30,37,33,40);(2,6)(4,7)(10,17)(14,24)(16,18)(19,22)(25,28,31,29)(26,33,35,30)(27,36,38,39)(32,37,34,40);(2,6)(3,8)(9,12,15,11)(10,18,17,16)(13,20,23,21)(14,22,24,19)(25,29,31,28)(26,33,35,30)(27,36,38,39)(32,40,34,37);(1,3)(2,7)(4,6)(5,8)(9,13,15,23)(10,19,17,22)(11,20,12,21)(14,18,24,16)(25,27)(26,34)(28,36)(29,39)(30,37)(31,38)(32,35)(33,40);(1,4,5,7)(2,3,6,8)(9,14,15,24)(10,20,17,21)(11,19,12,22)(13,18,23,16)(25,30,31,33)(26,28,35,29)(27,37,38,40)(32,39,34,36)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8306095);