Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| 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×#3054 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,20)(17,19)(21,25)(22,26)(23,27)(24,28);(2,3)(5,7,8,11)(6,9,10,12)(13,15)(14,18)(16,21)(17,23)(19,27)(20,25)(22,24)(26,28);(1,2,4,3)(6,10)(9,12)(15,21)(16,24)(18,25)(20,28)(22,23)(26,27);(1,3,4,2)(15,22)(16,24)(18,26)(20,28)(21,23)(25,27);(14,19)(18,27)(20,28)(25,26)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348822);