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 | |
4716 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)(3,4)(5,6)(7,8)(9,10)(11,16)(12,13)(14,22)(15,17)(18,23)(19,21)(20,24)(27,33)(31,38)(34,36)(39,40);(2,6)(3,7)(9,11,15,12)(10,16,17,13)(14,19,23,20)(18,24,22,21)(25,26)(27,31)(28,35)(29,30)(32,37)(33,38)(34,40)(36,39);(1,3,5,7)(2,4,6,8)(10,17)(11,12)(14,23)(21,24)(26,30)(27,33)(32,35)(34,36);(1,4)(2,7)(3,6)(5,8)(9,12,15,11)(10,16,17,13)(14,20,23,19)(18,24,22,21)(25,27)(26,31)(28,36)(29,33)(30,38)(32,40)(34,37)(35,39);(9,13,15,16)(10,12,17,11)(14,21,23,24)(18,20,22,19)(25,28,29,37)(26,32,30,35)(27,34,33,36)(31,39,38,40);(9,14)(10,18)(11,19)(12,20)(13,21)(15,23)(16,24)(17,22)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,37)(36,40)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8838246);