Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994107 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(7,11)(10,12)(13,14)(15,22)(16,17)(18,28)(19,21)(20,25)(23,24)(26,27);(6,9)(7,11)(13,15,19,26)(14,20,21,24)(16,25,18,23)(17,27,28,22);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,18)(14,17,21,28)(15,23,26,25)(20,22,24,27);(6,9)(10,12)(13,17,19,28)(14,16,21,18)(15,24,26,20)(22,23,27,25);(5,7,8,11)(6,10,9,12)(13,16,19,18)(14,17,21,28)(15,23,26,25)(20,22,24,27)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7055553);