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 | #59037 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,7,5,4)(9,10,14,18)(11,15,19,22)(12,16,21,24)(13,17,23,20)(26,31)(27,35)(28,38)(34,39);(1,3,6,5)(2,7,8,4)(9,11)(10,15)(12,20)(13,16)(14,19)(17,21)(18,22)(23,24)(25,26,30,31)(27,36,35,32)(28,33,38,40)(29,39,37,34);(2,8)(3,5)(11,19)(12,21)(15,22)(16,24)(25,27,30,35)(26,32,31,36)(28,37,38,29)(33,39,40,34);(1,3,6,5)(2,7,8,4)(9,12)(10,16)(11,20)(13,15)(14,21)(17,19)(18,24)(22,23)(25,28)(26,33)(27,37)(29,35)(30,38)(31,40)(32,39)(34,36);(1,4,6,7)(2,3,8,5)(9,13)(10,17)(11,16)(12,22)(14,23)(15,21)(18,20)(19,24)(25,29,30,37)(26,34,31,39)(27,28,35,38)(32,33,36,40);(1,5,6,3)(2,7,8,4)(9,11)(10,15)(12,20)(13,16)(14,19)(17,21)(18,22)(23,24)(25,28)(26,33)(27,37)(29,35)(30,38)(31,40)(32,39)(34,36)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=g,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815217);