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 |
| prime radical | #59410 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,15)(11,18)(12,20)(21,23)(25,26,31,30)(27,34,29,32)(28,33,38,37)(35,40,36,39);(1,2)(3,8)(4,6)(5,7)(9,10,14,15)(11,13,18,16)(12,17,20,22)(19,21,24,23)(25,27,31,29)(26,32,30,34)(28,36,38,35)(33,40,37,39);(1,3,5,6)(2,4,7,8)(9,11)(10,13)(12,19)(14,18)(15,16)(17,21)(20,24)(22,23)(25,28,31,38)(26,33,30,37)(27,35,29,36)(32,39,34,40);(1,2)(3,4)(5,7)(6,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,24)(21,22)(25,29,31,27)(26,34,30,32)(28,36,38,35)(33,40,37,39);(9,12,14,20)(10,17,15,22)(11,19,18,24)(13,21,16,23);(1,4)(2,6)(3,7)(5,8)(9,13,14,16)(10,18,15,11)(12,21,20,23)(17,24,22,19)(25,30,31,26)(27,34,29,32)(28,37,38,33)(35,40,36,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=i,d^2=gh,e^2=h,f^2=g,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, 8331742);