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 | #42674 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4380 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,17)(13,16)(15,20)(18,22)(19,24)(21,23)(25,26,29,32)(27,33,35,30)(28,36,38,31)(34,39,37,40);(1,3,5,4)(2,6,8,7)(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23)(25,27,29,35)(26,30,32,33)(28,37,38,34)(31,40,36,39);(1,3,5,4)(2,7,8,6)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,28,29,38)(26,31,32,36)(27,34,35,37)(30,39,33,40);(10,11)(15,17)(16,18)(23,24)(26,32)(30,33)(31,36)(39,40);(1,3,5,4)(2,7,8,6)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,e^2=gi,f^2=gh,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, 7548557);