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 | #52169 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4383 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)(4,8)(5,6)(9,10)(11,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,26)(27,30)(28,35)(29,32)(31,37)(33,40)(34,39)(36,38);(2,6)(7,8)(9,11,15,10)(12,19,22,16)(13,17,14,18)(20,23,21,24)(25,27,29,34)(26,30,32,39)(28,36,37,33)(31,40,35,38);(9,12,15,22)(10,16,11,19)(13,20,14,21)(17,23,18,24)(25,28,29,37)(26,31,32,35)(27,33,34,36)(30,38,39,40);(1,3,5,4)(2,7,6,8)(9,13,15,14)(10,17,11,18)(12,20,22,21)(16,23,19,24)(26,32)(30,39)(31,35)(38,40);(1,4,5,3)(2,8,6,7)(9,13,15,14)(10,18,11,17)(12,21,22,20)(16,23,19,24);(9,14,15,13)(10,17,11,18)(12,21,22,20)(16,23,19,24)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=g,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, 7548964);