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 | |
4556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,16)(12,13)(14,18)(15,19)(17,22)(20,23)(21,24)(26,31)(27,35)(28,34)(37,38);(1,2)(3,8)(4,6)(5,7)(11,19)(15,16)(20,24)(21,23)(25,26,30,31)(27,32,35,33)(28,29,34,39)(36,37,40,38);(1,3,5,6)(2,4,7,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,18)(17,23)(22,24)(25,27)(26,32)(28,36)(29,37)(30,35)(31,33)(34,40)(38,39);(1,2)(3,4)(5,7)(6,8)(9,11)(10,16)(12,20)(13,23)(14,19)(15,18)(17,21)(22,24)(25,27)(26,33)(28,36)(29,38)(30,35)(31,32)(34,40)(37,39);(9,12)(10,17)(11,20)(13,18)(14,22)(15,23)(16,21)(19,24)(25,28,30,34)(26,29,31,39)(27,36,35,40)(32,37,33,38);(1,4)(2,6)(3,7)(5,8)(9,13,14,17)(10,12,18,22)(11,21,19,23)(15,20,16,24)(25,29)(26,34)(27,37)(28,31)(30,39)(32,40)(33,36)(35,38)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8331974);