Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62812 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,9,13,10,14,8,11)(7,12,17,20,15,19,16,18);(6,11,14,13)(12,18,19,20);(1,2,3,4);(5,7)(6,12)(8,16)(9,17)(10,15)(11,18)(13,20)(14,19);(7,15)(12,19)(16,17)(18,20)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72267950 |
| maximal normal subgroup of | #72267950 |
| maximal quotient of | #72267953 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279160);