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 | #42570 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,16,19)(11,20,15,17)(12,14,13,18)(21,23,22,24)(25,26,30,34)(27,33,29,31)(28,32,36,39)(35,40,37,38);(2,3)(6,7)(9,11,16,15)(10,17,19,20)(12,22,13,21)(14,23,18,24)(25,27,30,29)(26,31,34,33)(28,35,36,37)(32,38,39,40);(1,3,5,2)(4,7,8,6)(9,12,16,13)(10,18,19,14)(11,21,15,22)(17,23,20,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(1,4,5,8)(2,6,3,7)(9,13,16,12)(10,14,19,18)(11,22,15,21)(17,24,20,23)(25,29,30,27)(26,33,34,31)(28,35,36,37)(32,38,39,40);(9,14)(10,12)(11,23)(13,19)(15,24)(16,18)(17,22)(20,21)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,40)(34,39);(9,15,16,11)(10,17,19,20)(12,22,13,21)(14,24,18,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=i,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8065962);