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 | #42011 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,15)(13,21)(14,16)(18,22)(19,23)(20,24)(25,26)(27,28)(29,30)(31,33)(32,34)(35,36)(37,38)(39,40);(1,2)(3,8)(4,7)(5,6)(9,11,14,12)(10,15,16,17)(13,20,22,19)(18,24,21,23)(25,27)(26,29)(28,34)(30,32)(31,35)(33,37)(36,40)(38,39);(2,6)(4,8)(10,16)(15,17)(18,21)(23,24)(25,28,32,29)(26,27,34,30)(31,37,39,36)(33,38,40,35);(2,6)(3,7)(9,12,14,11)(10,17,16,15)(13,19,22,20)(18,23,21,24)(25,29,32,28)(26,27,34,30)(31,37,39,36)(33,35,40,38);(1,3,5,7)(2,4,6,8)(9,13,14,22)(10,18,16,21)(11,19,12,20)(15,23,17,24)(25,30,32,27)(26,29,34,28)(31,38,39,35)(33,37,40,36);(1,4)(2,7)(3,6)(5,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,24)(21,22)(25,31)(26,33)(27,35)(28,36)(29,37)(30,38)(32,39)(34,40)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8302482);