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 | #2403 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,19)(11,20,23,16)(12,22,21,17)(13,18,24,14)(25,26,30,34)(27,35,36,31)(28,32,38,39)(29,33,37,40);(1,2,5,6)(3,7,8,4)(9,11,15,23)(10,16,19,20)(12,24,21,13)(14,22,18,17)(27,36)(28,38)(31,35)(32,39);(1,2)(3,7)(4,8)(5,6)(12,21)(13,24)(14,18)(17,22)(25,27,30,36)(26,31,34,35)(28,29,38,37)(32,40,39,33);(2,6)(3,8)(9,12)(10,17)(11,13)(14,20)(15,21)(16,18)(19,22)(23,24)(25,28,30,38)(26,32,34,39)(27,37,36,29)(31,33,35,40);(1,3)(2,4)(5,8)(6,7)(9,13)(10,18)(11,21)(12,23)(14,19)(15,24)(16,22)(17,20)(25,29,30,37)(26,33,34,40)(27,28,36,38)(31,39,35,32);(1,4)(2,3)(5,7)(6,8)(9,14,15,18)(10,13,19,24)(11,22,23,17)(12,20,21,16)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,40)(38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=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, 8968656);