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 | |
4522 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)(11,18)(12,21)(13,16)(14,22)(15,17)(19,20)(23,24)(25,26)(27,29)(28,31)(30,36)(32,34)(33,38)(35,39)(37,40);(2,6)(3,7)(9,11,15,13)(10,16,17,18)(12,19,22,23)(14,20,21,24)(25,27)(26,32)(28,29)(30,35)(31,34)(33,39)(36,37)(38,40);(1,3,5,7)(2,4,6,8)(10,17)(12,22)(16,18)(19,23)(26,28)(27,34)(33,36)(35,40);(1,4)(2,7)(3,6)(5,8)(9,12)(10,14)(11,19)(13,23)(15,22)(16,20)(17,21)(18,24)(25,28,31,26)(27,29,34,32)(30,36,38,33)(35,37,40,39);(9,13,15,11)(10,16,17,18)(12,23,22,19)(14,20,21,24)(25,29)(26,27)(28,34)(30,37)(31,32)(33,35)(36,40)(38,39);(9,14)(10,12)(11,20)(13,24)(15,21)(16,19)(17,22)(18,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,d^2=ghi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9771117);