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 | |
4520 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)(9,10,14,17)(11,18,21,15)(12,13,16,23)(19,20,22,24)(25,26)(27,32)(28,38)(29,40)(30,31)(33,36)(34,39)(35,37);(1,2,4,6)(3,5,8,7)(9,11,14,21)(10,15,17,18)(12,22,16,19)(13,20,23,24)(25,27)(26,32)(28,39)(29,36)(30,37)(31,35)(33,40)(34,38);(1,2)(3,5)(4,6)(7,8)(12,16)(13,23)(19,22)(20,24)(26,30)(27,35)(28,36)(34,40);(1,3,4,8)(2,5,6,7)(9,12,14,16)(10,13,17,23)(11,19,21,22)(15,24,18,20)(25,28)(26,33)(27,29)(30,38)(31,36)(32,34)(35,39)(37,40);(9,11)(10,15)(12,22)(13,20)(14,21)(16,19)(17,18)(23,24)(25,29)(26,34)(27,36)(28,35)(30,40)(31,39)(32,38)(33,37);(9,13)(10,16)(11,20)(12,17)(14,23)(15,19)(18,22)(21,24)(25,30,31,26)(27,37,35,32)(28,38,36,33)(29,40,39,34)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,d^2=g,f^2=h,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, 8841541);