Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C22×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,10)(8,9)(11,12)(13,14,16,19,17,20,15,18);(1,2,3,4)(6,9)(10,12);(14,18,20,19);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi176 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#21415, #72239521, #72239522 |
| maximal normal subgroup of | C4×#21415, #72239521, #72239522 |
| maximal quotient of | C4×#10923 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261363);