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 | #40912 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4496 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)(6,8)(9,10,14,12)(11,15,17,19)(13,16,22,20)(18,23,21,24)(25,26,31,34)(27,35,36,30)(28,32,37,40)(29,38,39,33);(9,11)(10,15)(12,19)(13,21)(14,17)(16,24)(18,22)(20,23)(25,27,31,36)(26,30,34,35)(28,29,37,39)(32,33,40,38);(1,2)(3,6)(4,5)(7,8)(11,17)(15,19)(18,21)(23,24)(25,28,31,37)(26,32,34,40)(27,29,36,39)(30,33,35,38);(1,3,4,7)(2,6,5,8)(9,12,14,10)(11,15,17,19)(13,20,22,16)(18,23,21,24)(25,29,31,39)(26,33,34,38)(27,28,36,37)(30,32,35,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,24)(21,22)(25,28)(26,32)(27,29)(30,33)(31,37)(34,40)(35,38)(36,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,30)(26,27)(28,33)(29,32)(31,35)(34,36)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=ghi,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, 10259734);