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 | #2615 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,20)(12,22)(13,14)(15,17)(16,21)(18,24)(19,23)(25,26,31,33)(27,32,28,36)(29,30,39,40)(34,38,37,35);(1,2,5,4)(3,6,7,8)(9,11,15,21)(10,16,17,20)(12,23,24,13)(14,18,19,22)(25,27,31,28)(26,32,33,36)(29,37,39,34)(30,35,40,38);(2,4)(6,8)(10,17)(14,19)(16,20)(18,22)(25,28,31,27)(26,32,33,36)(29,37,39,34)(30,38,40,35);(3,7)(6,8)(9,12,15,24)(10,18,17,22)(11,13,21,23)(14,20,19,16)(25,29)(26,30)(27,34)(28,37)(31,39)(32,38)(33,40)(35,36);(1,3)(2,6)(4,8)(5,7)(9,13)(10,19)(11,12)(14,17)(15,23)(16,22)(18,20)(21,24)(25,30)(26,29)(27,35)(28,38)(31,40)(32,37)(33,39)(34,36);(1,4,5,2)(3,8,7,6)(9,14,15,19)(10,13,17,23)(11,18,21,22)(12,20,24,16)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8232054);