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 | |
4568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,13)(12,20)(14,15)(16,18)(17,21)(19,24)(22,23)(25,26,31,35)(27,32,36,39)(28,29,33,40)(30,34,37,38);(1,3,4,7)(2,5,6,8)(10,15)(11,18)(12,21)(23,24)(25,27,31,36)(26,32,35,39)(28,37,33,30)(29,38,40,34);(1,2)(3,5)(4,6)(7,8)(10,15)(13,16)(17,20)(23,24)(27,36)(30,37)(32,39)(34,38);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,22)(13,15)(14,18)(17,23)(19,21)(20,24)(25,28)(26,29)(27,37)(30,36)(31,33)(32,38)(34,39)(35,40);(1,2)(3,5)(4,6)(7,8)(9,12,14,21)(10,17,15,20)(11,19,18,22)(13,23,16,24)(25,29)(26,33)(27,34)(28,35)(30,39)(31,40)(32,37)(36,38);(9,13,14,16)(10,11,15,18)(12,23,21,24)(17,19,20,22)(25,30,31,37)(26,34,35,38)(27,28,36,33)(29,39,40,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=ghi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8282817);