Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #36684 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10,14,18)(11,19,22,15)(12,17,13,16)(20,23,21,24)(25,26)(27,30)(28,35)(29,31)(32,36)(33,38)(34,40)(37,39);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,17)(13,16)(14,18)(19,22)(20,24)(21,23)(25,27,29,33)(26,30,31,38)(28,34,36,39)(32,37,35,40);(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19)(26,31)(27,33)(32,35)(34,39);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,19,24)(25,28,29,36)(26,32,31,35)(27,34,33,39)(30,37,38,40);(1,4,5,3)(2,6,7,8)(9,12,14,13)(10,17,18,16)(11,21,22,20)(15,23,19,24);(9,13,14,12)(10,16,18,17)(11,21,22,20)(15,23,19,24)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=h,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549000);