Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C127 ⋊ #435 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×D127 |
| commutator, socle | C22×C127 | C22×C4 |
| Fitting | C4×D4×C127 | C2 |
| hypercenter | C4×D4 | D127 |
| hypocenter | C127 | #435 |
| LCS | G ▷ C22×C127 ▷ C127 | |
| UCS | C1 ◁ C2×C4 ◁ C4×D4 | |
124 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^6,127|a^2=e,c^2=e,e^2=f,b^a=bd,c^a=cf,g^a=g^126,g^c=g^1261288 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #435×D127, C2×#22549948, #30647419, … |
| maximal normal subgroup of | #435×D127, C2×#22549948, #30647419, … |
| maximal quotient of | #30637560, #30637567, #30637574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 621);