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 | #40498 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,24)(20,22)(21,23)(25,26,32,35)(27,33,37,39)(28,38,29,34)(30,31,40,36);(2,6)(3,7)(10,14)(11,18)(16,22)(19,21)(25,27,32,37)(26,33,35,39)(28,36,29,31)(30,34,40,38);(1,3)(2,4)(5,7)(6,8)(12,20)(16,22)(19,21)(23,24)(25,28,32,29)(26,34,35,38)(27,36,37,31)(30,33,40,39);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,21)(13,18)(14,17)(16,24)(19,20)(22,23)(25,29,32,28)(26,34,35,38)(27,31,37,36)(30,33,40,39);(9,12)(10,16)(11,19)(13,20)(14,22)(15,23)(17,24)(18,21)(25,30)(26,31)(27,34)(28,33)(29,39)(32,40)(35,36)(37,38);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,31)(26,30)(27,28)(29,37)(32,36)(33,34)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10232539);