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 | |
4558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,16)(11,12)(13,21)(22,24)(26,32)(28,33)(34,40)(36,38);(1,2)(3,8)(4,6)(5,7)(9,10)(11,17)(12,14)(13,18)(15,16)(19,24)(20,22)(21,23)(25,26)(27,28)(29,33)(30,34)(31,32)(35,36)(37,38)(39,40);(1,3,5,6)(2,4,7,8)(9,11,15,12)(10,17,16,14)(13,19,21,20)(18,24,23,22)(25,27,31,29)(26,28,32,33)(30,35,39,37)(34,36,40,38);(1,2)(3,4)(5,7)(6,8)(9,12,15,11)(10,17,16,14)(13,20,21,19)(18,24,23,22)(25,28,31,33)(26,27,32,29)(30,38,39,36)(34,37,40,35);(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,29,31,27)(26,33,32,28)(30,37,39,35)(34,38,40,36);(1,4)(2,6)(3,7)(5,8)(9,14,15,17)(10,12,16,11)(13,22,21,24)(18,20,23,19)(25,30)(26,34)(27,35)(28,36)(29,37)(31,39)(32,40)(33,38)Polycyclic presentation:
2^9|c^2=gi,d^2=g,e^2=gh,f^2=h,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, 8332115);