Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,9,11,12)(8,10)(13,14,16,19,17,15,18,20);(2,4)(5,7)(6,9,10,12)(8,11);(5,7)(6,9,10,12)(8,11)(13,15,16,20,17,14,18,19)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72258158, #72258168, #72258178 |
| maximal normal subgroup of | #72258158, #72258168, #72258178 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17270);