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 | #35707 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4333 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)(4,7)(5,6)(25,26,30,32)(27,31,35,40)(28,36,38,29)(33,34,37,39);(1,2,6,5)(3,7,4,8)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(25,26)(27,31)(28,29)(30,32)(33,39)(34,37)(35,40)(36,38);(1,3)(2,7)(4,6)(5,8)(10,15)(12,19)(13,21)(18,23)(25,27,30,35)(26,31,32,40)(28,33,38,37)(29,39,36,34);(1,2,6,5)(3,7,4,8)(9,11)(10,12)(13,18)(14,16)(15,19)(17,20)(21,23)(22,24)(25,28)(26,29)(27,33)(30,38)(31,39)(32,36)(34,40)(35,37);(1,4,6,3)(2,7,5,8)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,27)(26,31)(28,37)(29,34)(30,35)(32,40)(33,38)(36,39);(1,5,6,2)(3,7,4,8)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,29)(26,28)(27,34)(30,36)(31,37)(32,38)(33,40)(35,39)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8849815);