Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2359 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)(5,6,8,9)(7,10,11,12)(13,14)(15,21)(16,17)(18,20)(19,24)(22,23)(25,28)(26,27);(6,9)(10,12)(13,15)(14,19)(16,21)(17,24)(18,23)(20,28)(22,26)(25,27);(2,4)(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,22,24,28)(18,20,27,26)(19,23,21,25);(15,23)(18,27)(19,28)(20,26)(21,22)(24,25);(14,16)(15,24)(20,26)(23,25)Polycyclic presentation:
2^9|a^2=fi,c^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72315711, #72315744, #72315755 |
| maximal normal subgroup of | #72315711, #72315744, #72315755 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345682);