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 | |
4333 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)(3,5)(4,7)(6,8)(9,10,14,16)(11,15,17,12)(13,18,22,23)(19,21,24,20)(27,33)(32,39)(34,37)(35,40);(1,2,6,8)(3,5,7,4)(9,11,14,17)(10,15,16,12)(13,19,22,24)(18,21,23,20)(25,26,30,31)(27,32,33,39)(28,36,38,29)(34,35,37,40);(1,3,6,7)(2,4,8,5)(10,16)(11,17)(18,23)(19,24)(26,31)(29,36)(32,39)(35,40);(1,4,6,5)(2,3,8,7)(9,12,14,15)(10,17,16,11)(13,21,22,20)(18,19,23,24)(25,27)(26,32)(28,37)(29,35)(30,33)(31,39)(34,38)(36,40);(1,2)(3,4)(5,7)(6,8)(9,13,14,22)(10,18,16,23)(11,19,17,24)(12,20,15,21)(25,28)(26,29)(27,34)(30,38)(31,36)(32,40)(33,37)(35,39);(1,5,6,4)(2,7,8,3)(9,10)(11,15)(12,17)(13,18)(14,16)(19,21)(20,24)(22,23)(25,29)(26,28)(27,35)(30,36)(31,38)(32,37)(33,40)(34,39)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=g,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833597);