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 | #2847 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,8)(6,7)(9,10)(11,17)(12,15)(13,16)(14,18)(19,23)(20,22)(21,24)(25,26,31,29)(27,34,33,30)(28,36,38,32)(35,37,39,40);(1,3)(2,8)(4,6)(5,7)(9,10,13,16)(11,14,18,17)(12,15,20,22)(19,21,24,23)(26,29)(27,33)(32,36)(35,39);(1,2,7,6)(3,8,5,4)(11,18)(12,20)(14,17)(15,22)(25,27)(26,30)(28,35)(29,34)(31,33)(32,37)(36,40)(38,39);(1,4)(2,5)(3,6)(7,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,17)(20,24)(22,23)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40);(1,5,7,3)(2,8,6,4)(9,12)(10,15)(11,19)(13,20)(14,21)(16,22)(17,23)(18,24)(25,29,31,26)(27,34,33,30)(28,36,38,32)(35,40,39,37);(1,6,7,2)(3,8,5,4)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,30,31,34)(26,33,29,27)(28,37,38,40)(32,39,36,35)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8679679);