Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×D4×#41 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4619 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)(7,11)(8,10)(9,12)(13,14,17,20,16,18,24,25)(15,21,23,28,22,19,26,27);(5,7)(6,9)(8,12)(10,11)(14,18)(19,21)(20,25)(27,28);(2,4)(7,12)(9,11);(6,10)(9,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,27)(24,26)(25,28);(15,22)(19,21)(23,26)(27,28)Polycyclic presentation:
2^9|a^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281427);