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 | #36440 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,15,17)(11,16,12,18)(13,21,23,19)(14,22,24,20)(28,36)(31,38)(34,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,15)(13,20)(14,21)(17,18)(19,24)(22,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,2)(3,6)(4,5)(7,8)(10,17)(11,12)(19,21)(20,22)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(2,5)(6,7)(9,12,15,11)(10,18,17,16)(13,20,23,22)(14,19,24,21)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,23)(10,19,17,21)(11,20,12,22)(14,16,24,18)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,14)(10,20)(11,21)(12,19)(13,16)(15,24)(17,22)(18,23)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064551);