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 | #46978 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,16)(11,15,20,19)(12,17,21,22)(13,18,23,24)(25,26,30,33)(27,31,35,29)(28,36,38,32)(34,37,40,39);(1,2,5,4)(3,7,8,6)(9,11)(10,15)(12,18)(13,22)(14,20)(16,19)(17,23)(21,24)(25,27,30,35)(26,29,33,31)(28,34,38,40)(32,37,36,39);(1,3,5,8)(2,6,4,7)(10,16)(11,20)(12,21)(13,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(1,3,5,8)(2,7,4,6)(9,12,14,21)(10,17,16,22)(11,18,20,24)(13,15,23,19)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,37,31,39);(1,3)(2,7)(4,6)(5,8)(9,13)(10,18)(11,17)(12,19)(14,23)(15,21)(16,24)(20,22)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,34,36,40);(1,4,5,2)(3,6,8,7)(9,11)(10,19)(12,18)(13,17)(14,20)(15,16)(21,24)(22,23)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=i,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9514827);