Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| socle | C23 | C2×#99 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1034 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,8,11,9,12,7,10)(13,14)(15,18,20,17)(16,19);(1,2,3,4)(14,17,19,18);(5,7,9,8)(6,10,12,11)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,d^2=i,e^2=h,b^a=bd,d^a=dg,f^a=fi,g^a=gi,e^b=egi,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72282612, #72282624, #72282658, … |
| maximal normal subgroup of | #72282612, #72282624, #72282658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34147);