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 | |
4451 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)(6,8)(9,10,14,15)(11,18,17,13)(12,16,20,22)(19,24,23,21)(25,26,30,32)(27,31,35,40)(28,33,34,29)(36,39,38,37);(3,7)(6,8)(10,15)(12,20)(13,18)(19,23)(25,27)(26,31)(28,38)(29,39)(30,35)(32,40)(33,37)(34,36);(1,2,5,4)(3,6,7,8)(10,15)(11,17)(12,20)(21,24)(26,32)(27,35)(28,34)(37,39);(9,11,14,17)(10,13,15,18)(12,19,20,23)(16,21,22,24)(25,27)(26,31)(28,36)(29,37)(30,35)(32,40)(33,39)(34,38);(1,3,5,7)(2,6,4,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,28,30,34)(26,33,32,29)(27,36,35,38)(31,39,40,37);(1,4,5,2)(3,8,7,6)(9,13)(10,17)(11,15)(12,21)(14,18)(16,23)(19,22)(20,24)(25,29,30,33)(26,34,32,28)(27,37,35,39)(31,38,40,36)Polycyclic presentation:
2^9|a^2=gi,c^2=ghi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8679618);