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 | |
4447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,8,4,7)(9,10,15,18)(11,16,13,12)(14,17,23,22)(19,24,21,20)(25,26,30,32)(27,33,34,28)(29,31,38,40)(35,37,39,36);(2,6)(7,8)(9,11,15,13)(10,12,18,16)(14,19,23,21)(17,20,22,24)(27,34)(28,33)(35,39)(36,37);(1,3,5,4)(2,7,6,8)(9,12,15,16)(10,11,18,13)(14,20,23,24)(17,19,22,21)(25,27)(26,28)(29,35)(30,34)(31,36)(32,33)(37,40)(38,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,12)(13,16)(14,22)(15,18)(17,23)(19,24)(20,21)(25,28)(26,27)(29,37)(30,33)(31,39)(32,34)(35,40)(36,38);(9,13,15,11)(10,16,18,12)(14,21,23,19)(17,24,22,20);(1,4,5,3)(2,7,6,8)(9,14)(10,17)(11,19)(12,20)(13,21)(15,23)(16,24)(18,22)(25,29)(26,31)(27,35)(28,36)(30,38)(32,40)(33,37)(34,39)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8370856);