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 | #35226 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,17)(16,20)(22,23)(25,26,30,33)(27,31,35,29)(28,36,39,32)(34,38,37,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,27)(26,31)(28,37)(29,33)(30,35)(32,38)(34,39)(36,40);(2,3)(6,7)(11,18)(15,17)(19,24)(22,23)(25,28,30,39)(26,32,33,36)(27,34,35,37)(29,38,31,40);(4,8)(6,7)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,23)(22,24)(25,29)(26,27)(28,38)(30,31)(32,34)(33,35)(36,37)(39,40);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,28)(26,32)(27,34)(29,38)(30,39)(31,40)(33,36)(35,37);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,40)(36,38)(37,39)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8520087);