Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| 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×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1891 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,18)(17,21)(19,22)(20,27)(24,28)(25,26);(2,3)(13,15,18,27,19,28,17,26)(14,20,21,23,22,25,16,24);(1,2,4,3)(6,9)(10,12)(13,16,19,21)(14,17,22,18)(15,24,28,23)(20,26,25,27);(1,3,4,2)(13,14)(15,25)(16,17)(18,21)(19,22)(20,28)(23,26)(24,27);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,21,22,16)(15,26,28,27)(20,23,25,24)Polycyclic presentation:
2^9|b^2=f,c^2=hi,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375016);