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 | |
4391 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,21)(13,24)(15,17)(16,20)(18,23)(19,22)(25,26,29,32)(27,30,33,37)(28,35,36,31)(34,40,39,38);(2,6)(3,8)(9,11,15,20)(10,16,17,14)(12,22,23,13)(18,19,21,24)(27,33)(30,37)(34,39)(38,40);(1,3)(2,4)(5,8)(6,7)(10,17)(14,16)(18,21)(19,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,3)(2,4)(5,8)(6,7)(9,12,15,23)(10,18,17,21)(11,13,20,22)(14,19,16,24)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(1,4)(2,3)(5,7)(6,8)(9,13)(10,19)(11,12)(14,18)(15,22)(16,21)(17,24)(20,23)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40);(9,14)(10,11)(12,19)(13,18)(15,16)(17,20)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,d^2=gi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8202742);