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 | |
4518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,17)(13,18)(14,15)(16,20)(21,22)(23,24)(27,29)(31,34)(35,36)(39,40);(1,2)(3,8)(4,6)(5,7)(10,15)(11,20)(17,22)(18,24)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,39)(37,38);(1,3,5,6)(2,4,7,8)(25,27,30,29)(26,31,33,34)(28,35,38,36)(32,39,37,40);(1,4,5,8)(2,3,7,6)(9,11)(10,16)(12,18)(13,22)(14,20)(15,19)(17,23)(21,24)(25,28)(26,32)(27,35)(29,36)(30,38)(31,39)(33,37)(34,40);(1,2)(3,4)(5,7)(6,8)(9,12)(10,17)(11,18)(13,19)(14,21)(15,22)(16,23)(20,24)(25,29,30,27)(26,31,33,34)(28,36,38,35)(32,39,37,40);(1,4)(2,6)(3,7)(5,8)(9,13,14,23)(10,18,15,24)(11,17,20,22)(12,16,21,19)(25,26)(27,31)(28,37)(29,34)(30,33)(32,38)(35,40)(36,39)Polycyclic presentation:
2^9|c^2=i,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8604800);