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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,4)(12,20)(16,21)(18,24)(22,23)(25,26)(27,32)(28,37)(29,39)(30,35)(31,36)(33,38)(34,40);(1,2,6,5)(3,8,4,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(27,36)(28,38)(31,32)(33,37);(1,3,6,4)(2,7,5,8)(10,14)(15,17)(16,21)(22,23)(25,27,30,36)(26,31,35,32)(28,34,38,39)(29,33,40,37);(1,4,6,3)(2,8,5,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27)(26,32)(28,34)(29,33)(30,36)(31,35)(37,40)(38,39);(1,5,6,2)(3,8,4,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28)(26,33)(27,34)(29,32)(30,38)(31,40)(35,37)(36,39);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,34)(27,33)(28,32)(30,40)(31,38)(35,39)(36,37)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017923);