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 | |
4456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(7,8)(9,10,15,16)(11,17,13,12)(14,22,23,18)(19,20,21,24)(25,26,30,32)(27,31,35,40)(28,36,38,33)(29,39,37,34);(1,2)(3,8)(4,6)(5,7)(9,11,15,13)(10,12,16,17)(14,19,23,21)(18,20,22,24)(25,27,30,35)(26,31,32,40)(28,37,38,29)(33,39,36,34);(3,5)(7,8)(10,16)(12,17)(14,23)(19,21)(26,32)(27,35)(28,38)(34,39);(1,3,6,5)(2,7,4,8)(9,12)(10,11)(13,16)(14,20)(15,17)(18,19)(21,22)(23,24)(25,28,30,38)(26,33,32,36)(27,29,35,37)(31,34,40,39);(1,4,6,2)(3,7,5,8)(9,13,15,11)(10,17,16,12)(14,21,23,19)(18,24,22,20)(25,29)(26,34)(27,28)(30,37)(31,33)(32,39)(35,38)(36,40);(1,5,6,3)(2,7,4,8)(9,14,15,23)(10,18,16,22)(11,19,13,21)(12,20,17,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,40)(36,38)(37,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8840379);