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 | #43011 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10,15,19)(11,20,23,16)(12,22,21,17)(13,18,24,14);(1,2,6,4)(3,8,7,5)(9,11,15,23)(10,16,19,20)(12,24,21,13)(14,22,18,17)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,39)(38,40);(1,3,6,7)(2,5,4,8)(12,21)(13,24)(14,18)(17,22)(26,30)(27,34)(28,37)(39,40);(2,4)(3,7)(9,12)(10,17)(11,13)(14,20)(15,21)(16,18)(19,22)(23,24)(25,27)(26,31)(28,38)(29,34)(30,33)(32,40)(35,37)(36,39);(1,4,6,2)(3,5,7,8)(9,13)(10,18)(11,21)(12,23)(14,19)(15,24)(16,22)(17,20)(25,28)(26,32)(27,35)(29,37)(30,36)(31,39)(33,40)(34,38);(1,5)(2,3)(4,7)(6,8)(9,14,15,18)(10,13,19,24)(11,22,23,17)(12,20,21,16)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,37)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^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, 8966178);