Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,9,10)(7,11,8,12)(13,14)(15,21)(16,25)(17,23)(18,20)(19,22)(24,27)(26,28);(6,10)(11,12)(14,20)(15,17)(16,19)(21,28)(23,27)(24,26);(2,4)(13,15)(14,21)(16,17)(18,24)(19,26)(20,27)(22,28)(23,25);(5,7,9,8)(6,11,10,12)(13,16)(14,22)(17,26)(18,19)(20,25)(21,27);(5,8,9,7)(6,12,10,11)(13,17,18,26)(14,23,20,28)(15,16,24,19)(21,25,27,22)Polycyclic presentation:
2^9|a^2=fi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485374);