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 | #46529 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4259 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)(11,18)(12,13)(14,15)(16,19)(17,22)(20,24)(21,23)(25,26)(27,32)(28,35)(29,33)(30,31)(34,39)(36,40)(37,38);(1,2,5,7)(3,4,8,6)(10,15)(11,19)(13,22)(20,23)(26,31)(28,32)(29,37)(34,40);(1,3,5,8)(2,6,7,4)(10,15)(12,17)(16,18)(20,23)(25,27,30,35)(26,32,31,28)(29,34,37,40)(33,39,38,36);(1,3,5,8)(2,6,7,4)(9,11,14,19)(10,16,15,18)(12,20,17,23)(13,21,22,24)(25,28)(26,27)(29,36)(30,32)(31,35)(33,34)(37,39)(38,40);(1,3)(2,4)(5,8)(6,7)(9,12,14,17)(10,13,15,22)(11,20,19,23)(16,21,18,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,4)(2,3)(5,6)(7,8)(9,13)(10,17)(11,21)(12,15)(14,22)(16,23)(18,20)(19,24)(25,29,30,37)(26,33,31,38)(27,34,35,40)(28,36,32,39)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10275652);