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 | #40492 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(4,8)(26,30)(27,33)(28,36)(37,40);(2,6)(3,7)(9,10,14,15)(11,16,19,24)(12,17,22,21)(13,18,23,20)(25,26,31,30)(27,34,33,29)(28,32,36,38)(35,40,39,37);(1,2,5,6)(3,8,7,4)(10,15)(11,19)(12,22)(13,23)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,39)(38,40);(2,6)(3,7)(9,11)(10,16)(12,18)(13,21)(14,19)(15,24)(17,23)(20,22)(25,28)(26,32)(27,35)(29,37)(30,38)(31,36)(33,39)(34,40);(1,3,5,7)(2,4,6,8)(9,12)(10,17)(11,20)(13,16)(14,22)(15,21)(18,19)(23,24)(25,29,31,34)(26,33,30,27)(28,37,36,40)(32,39,38,35);(1,4)(2,7)(3,6)(5,8)(9,13)(10,18)(11,21)(12,16)(14,23)(15,20)(17,19)(22,24)(25,30,31,26)(27,34,33,29)(28,38,36,32)(35,40,39,37)Polycyclic presentation:
2^9|b^2=ghi,c^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8884598);