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 | |
4618 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)(6,7)(9,10)(11,17)(12,16)(13,22)(14,19)(15,20)(18,23)(21,24)(25,26,30,31)(27,28,35,32)(29,33,37,38)(34,36,40,39);(1,2,5,4)(3,7,8,6)(9,11,15,12)(10,16,20,17)(13,19,23,21)(14,22,24,18)(26,31)(28,32)(29,37)(34,40);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,17,20,16)(13,21,23,19)(14,22,24,18)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,17,20,16)(13,19,23,21)(14,22,24,18)(25,28,30,32)(26,27,31,35)(29,36,37,39)(33,34,38,40);(9,13)(10,18)(11,21)(12,19)(14,17)(15,23)(16,24)(20,22)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40);(9,14)(10,19)(11,22)(12,18)(13,17)(15,24)(16,23)(20,21)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8378849);