Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D8 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,8)(9,10,11,12)(14,21)(15,23)(16,27)(17,18)(19,28)(22,24)(25,26);(2,4)(13,14,18,23,19,15,16,21,20,24,27,25,28,26,17,22);(5,6,7,8)(13,15,18,21,19,24,16,25,20,26,27,22,28,14,17,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=eh,d^2=hi,e^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,h^a=hi,c^b=ci104 conjugacy classes — view the full character table.
| maximal subgroup of | #72210403 |
| maximal normal subgroup of | #72210403 |
| maximal quotient of | #72210358 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54286);