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 | |
4573 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)(3,7)(4,6)(5,8)(25,26)(27,31)(28,32)(29,35)(30,33)(34,39)(36,38)(37,40);(1,2,4,6)(3,5,8,7)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,23)(22,24)(25,27,30,35)(26,31,33,29)(28,34,38,40)(32,39,36,37);(1,2)(3,5)(4,6)(7,8)(10,14)(11,18)(12,21)(23,24)(25,26,30,33)(27,29,35,31)(28,36,38,32)(34,39,40,37);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,22)(13,18)(14,17)(16,24)(19,21)(20,23)(25,26)(27,29)(28,36)(30,33)(31,35)(32,38)(34,39)(37,40);(9,12)(10,16)(11,19)(13,21)(14,20)(15,23)(17,24)(18,22)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,36)(35,40);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,29)(26,27)(28,37)(30,31)(32,34)(33,35)(36,40)(38,39)Polycyclic presentation:
2^9|b^2=gh,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=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, 8965461);