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 | #58814 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4265 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)(11,20)(12,14)(13,23)(15,19)(16,22)(17,21)(18,24)(25,26,31,30)(27,34,33,29)(28,32,36,38)(35,40,39,37);(1,2,5,4)(3,6,7,8)(9,11,15,22)(10,16,19,20)(12,13,21,24)(14,18,17,23)(28,36)(32,38)(35,39)(37,40);(2,4)(6,8)(9,12,15,21)(10,17,19,14)(11,13,22,24)(16,23,20,18)(25,27)(26,29)(28,35)(30,34)(31,33)(32,37)(36,39)(38,40);(3,7)(6,8)(9,11)(10,16)(12,13)(14,18)(15,22)(17,23)(19,20)(21,24)(25,28,31,36)(26,32,30,38)(27,35,33,39)(29,37,34,40);(1,3)(2,6)(4,8)(5,7)(9,13)(10,18)(11,21)(12,22)(14,20)(15,24)(16,17)(19,23)(25,29,31,34)(26,33,30,27)(28,37,36,40)(32,39,38,35);(1,4,5,2)(3,8,7,6)(9,14,15,17)(10,12,19,21)(11,18,22,23)(13,20,24,16)(25,30,31,26)(27,34,33,29)(28,38,36,32)(35,40,39,37)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=g,d^2=gh,e^2=gh,f^2=hi,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, 8233198);