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 | |
4631 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,21)(22,24)(25,26,30,34)(27,33,37,31)(28,29,39,32)(35,40,38,36);(2,6)(3,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,22)(21,24)(25,27,30,37)(26,31,34,33)(28,35,39,38)(29,36,32,40);(1,3)(2,4)(5,8)(6,7)(10,14)(11,13)(12,20)(23,24)(25,28,30,39)(26,32,34,29)(27,35,37,38)(31,40,33,36);(1,4)(2,3)(5,7)(6,8)(9,12,15,20)(10,17,14,21)(11,19,13,22)(16,23,18,24)(25,28)(26,32)(27,35)(29,34)(30,39)(31,40)(33,36)(37,38);(1,2)(3,4)(5,6)(7,8)(9,13,15,11)(10,18,14,16)(12,19,20,22)(17,23,21,24)(25,27)(26,33)(28,38)(29,36)(30,37)(31,34)(32,40)(35,39);(1,4)(2,3)(5,7)(6,8)(9,14,15,10)(11,18,13,16)(12,21,20,17)(19,24,22,23)(25,29,30,32)(26,28,34,39)(27,36,37,40)(31,35,33,38)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=hi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8937348);