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 | |
4512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,14,16)(11,17,19,12)(13,15,21,22)(18,23,24,20);(2,3)(6,7)(9,10)(11,17)(12,19)(13,15)(14,16)(18,23)(20,24)(21,22)(25,26,30,31)(27,33,36,32)(28,37,39,29)(34,40,38,35);(1,2,5,3)(4,6,8,7)(13,21)(15,22)(18,24)(20,23)(26,31)(29,37)(32,33)(35,40);(1,3,5,2)(4,7,8,6)(9,11,14,19)(10,12,16,17)(13,18,21,24)(15,20,22,23)(25,27,30,36)(26,32,31,33)(28,38,39,34)(29,35,37,40);(1,4,5,8)(2,6,3,7)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,28)(26,29)(27,34)(30,39)(31,37)(32,40)(33,35)(36,38);(9,13,14,21)(10,15,16,22)(11,18,19,24)(12,20,17,23)(25,29)(26,28)(27,35)(30,37)(31,39)(32,38)(33,34)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=h,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8302233);