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 | |
4344 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)(3,8)(4,6)(5,7)(9,10)(11,18)(12,15)(13,20)(14,17)(16,22)(19,21)(23,24)(25,26,30,32)(27,33,34,28)(29,37,38,31)(35,36,40,39);(2,4)(7,8)(10,12)(11,17)(16,20)(19,23)(27,34)(28,33)(29,38)(31,37);(1,3,6,5)(2,7,4,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,27)(26,28)(29,35)(30,34)(31,36)(32,33)(37,39)(38,40);(1,2,6,4)(3,7,5,8)(9,12,15,10)(11,18,17,14)(13,20,22,16)(19,24,23,21)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(1,4,6,2)(3,8,5,7)(9,13,15,22)(10,16,12,20)(11,19,17,23)(14,21,18,24)(25,29,30,38)(26,31,32,37)(27,35,34,40)(28,36,33,39);(1,5,6,3)(2,8,4,7)(9,14,15,18)(10,17,12,11)(13,21,22,24)(16,23,20,19)(25,26)(27,28)(29,31)(30,32)(33,34)(35,36)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=gh,c^2=hi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015921);