Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| 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 | |
5127 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)(6,10,12,15)(11,17,18,20);(5,6,8,15,9,12,16,10)(7,11,14,20,13,18,19,17);(1,2,3,4);(5,7)(6,11)(8,14)(9,13)(10,17)(12,18)(15,20)(16,19);(7,13)(11,18)(14,19)(17,20)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,f^2=i,b^a=bf,c^a=cg,f^b=fi,h^b=hi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72344013 |
| maximal normal subgroup of | #72344013 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279233);