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 | #45789 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,8,4,5)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(2,7)(3,4)(9,10,14,15)(11,13,17,22)(12,19,21,16)(18,24,23,20)(26,30)(31,33)(32,37)(38,39);(2,7)(5,8)(10,15)(13,22)(16,19)(20,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,3,6,4)(2,5,7,8)(9,11,14,17)(10,13,15,22)(12,18,21,23)(16,20,19,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,4,6,3)(2,8,7,5)(9,12,14,21)(10,16,15,19)(11,18,17,23)(13,20,22,24);(1,5,6,8)(2,3,7,4)(9,13)(10,17)(11,15)(12,20)(14,22)(16,23)(18,19)(21,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8134563);