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 | |
4419 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)(11,18)(12,22)(13,20)(14,19)(15,21)(16,23)(17,24)(25,26,31,37)(27,32,39,40)(28,34,38,33)(29,30,36,35);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,20)(13,16)(14,21)(17,23)(18,19)(22,24)(25,27,31,39)(26,32,37,40)(28,30,38,35)(29,34,36,33);(1,3)(2,5)(4,8)(6,7)(9,12,14,23)(10,16,19,22)(11,20,21,17)(13,18,24,15)(25,28,31,38)(26,33,37,34)(27,35,39,30)(29,32,36,40);(3,8)(5,6)(9,12)(10,16)(11,17)(13,18)(14,23)(15,24)(19,22)(20,21)(25,28)(26,34)(27,35)(29,32)(30,39)(31,38)(33,37)(36,40);(1,2)(3,5)(4,7)(6,8)(9,13)(10,17)(11,16)(12,18)(14,24)(15,23)(19,20)(21,22)(25,29)(26,35)(27,34)(28,32)(30,37)(31,36)(33,39)(38,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,18)(12,20)(13,22)(14,21)(15,19)(16,24)(17,23)(25,30,31,35)(26,36,37,29)(27,38,39,28)(32,33,40,34)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8611322);