Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| 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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24)(25,26,29,32)(27,30,34,38)(28,35,36,31)(33,39,40,37);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(3,7)(5,8)(25,28,29,36)(26,31,32,35)(27,33,34,40)(30,37,38,39);(10,15)(14,17)(16,22)(21,23)(27,34)(30,38)(33,40)(37,39);(2,6)(3,7)(10,15)(11,18)(16,22)(19,24)(26,32)(30,38)(31,35)(37,39);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012924);