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 | |
4500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(11,17)(15,18)(19,23)(22,24)(25,26)(27,33)(28,31)(29,36)(30,34)(32,38)(35,39)(37,40);(3,7)(6,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,27,30,36)(26,29,34,33)(28,35,38,40)(31,37,32,39);(1,2,5,4)(3,6,7,8)(10,14)(15,18)(16,20)(22,24)(25,28,30,38)(26,31,34,32)(27,35,36,40)(29,37,33,39);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28)(26,32)(27,35)(29,39)(30,38)(31,34)(33,37)(36,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,29,30,33)(26,27,34,36)(28,37,38,39)(31,35,32,40);(1,4,5,2)(3,8,7,6)(9,12,13,21)(10,16,14,20)(11,19,17,23)(15,22,18,24)(25,27)(26,33)(28,35)(29,34)(30,36)(31,39)(32,37)(38,40)Polycyclic presentation:
2^9|b^2=g,c^2=gh,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8784291);