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 | #34589 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4403 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,20)(13,18)(14,17)(16,22)(19,23)(21,24)(25,26,30,29)(27,31,34,38)(28,32,37,36)(33,39,35,40);(2,5)(6,8)(9,11,14,18)(10,15,17,13)(12,19,22,24)(16,21,20,23)(25,27,30,34)(26,31,29,38)(28,35,37,33)(32,40,36,39);(9,12,14,22)(10,16,17,20)(11,19,18,24)(13,23,15,21)(25,28)(26,32)(27,33)(29,36)(30,37)(31,39)(34,35)(38,40);(10,17)(11,18)(16,20)(19,24)(25,29,30,26)(27,31,34,38)(28,36,37,32)(33,39,35,40);(9,13,14,15)(10,18,17,11)(12,21,22,23)(16,19,20,24)(28,37)(32,36)(33,35)(39,40);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,d^2=i,e^2=h,f^2=g,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, 7548081);