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 | #42619 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4371 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)(11,16)(12,18)(13,14)(15,17)(19,21)(20,23)(22,24)(25,26,32,38)(27,39,40,33)(28,35,29,34)(30,31,37,36);(1,2,5,4)(3,6,7,8)(9,11,15,14)(10,16,17,13)(12,20,22,19)(18,23,24,21)(25,27)(26,33)(28,37)(29,30)(31,35)(32,40)(34,36)(38,39);(2,4)(6,8)(10,17)(11,14)(18,24)(19,20)(25,28,32,29)(26,34,38,35)(27,37,40,30)(31,33,36,39);(3,7)(6,8)(9,12)(10,18)(11,19)(13,23)(14,20)(15,22)(16,21)(17,24)(25,29,32,28)(26,35,38,34)(27,37,40,30)(31,33,36,39);(1,3,5,7)(2,6,4,8)(9,13)(10,11)(12,21)(14,17)(15,16)(18,20)(19,24)(22,23)(25,30,32,37)(26,36,38,31)(27,29,40,28)(33,34,39,35);(1,4,5,2)(3,8,7,6)(9,14,15,11)(10,13,17,16)(12,20,22,19)(18,23,24,21)(25,31,32,36)(26,37,38,30)(27,35,40,34)(28,39,29,33)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=g,e^2=hi,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, 8249060);