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 | |
4511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(6,8)(9,10,14,16)(11,19,21,15)(12,22,23,17)(13,24,20,18)(25,26,30,31)(27,32,34,29)(28,36,38,33)(35,37,40,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,16,19)(12,20,23,13)(17,18,22,24)(26,31)(27,34)(33,36)(35,40);(2,5)(6,8)(10,16)(15,19)(17,22)(18,24)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,17)(13,18)(14,16)(15,21)(20,24)(22,23)(25,28,30,38)(26,33,31,36)(27,35,34,40)(29,37,32,39);(9,12)(10,17)(11,20)(13,21)(14,23)(15,18)(16,22)(19,24);(1,3)(2,6)(4,7)(5,8)(9,13,14,20)(10,18,16,24)(11,12,21,23)(15,22,19,17)(25,29,30,32)(26,27,31,34)(28,37,38,39)(33,35,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013254);