Finite Group DatabasebetaFinite Group Databasebeta| order | 8128 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C127 ⋊ #397 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D127 |
| commutator, socle | C23×C127 | C23 |
| Fitting | C2×C22⋊C4×C127 | C2 |
| hypercenter | C2×C22⋊C4 | D127 |
| hypocenter | C127 | #397 |
| LCS | G ▷ C23×C127 ▷ C127 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊C4 | |
170 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 170 conjugacy classes of subgroups — too many to draw. Only its 59 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=d,b^a=bd,c^a=ce,c^b=cf,g^c=g^1261282 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #30636729, #30636731, #30637716, … |
| maximal normal subgroup of | #30636729, #30636731, #30637716, … |
| maximal quotient of | #30636975, #30636977, #30636986, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8128, 474);