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×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,21)(17,20)(18,22)(23,28)(24,26)(25,27);(6,9)(10,12)(13,15,17,25)(14,19,20,27)(16,18,23,24)(21,22,28,26);(1,2,3,4)(14,20)(15,23)(16,25)(18,24)(19,21)(27,28);(5,7,8,11)(6,10,9,12)(13,16,17,23)(14,21,20,28)(15,24,25,18)(19,26,27,22);(15,16)(18,24)(19,21)(22,26)(23,25)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=fi,h^2=i,b^a=bf,c^a=cg,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, 6491058);