Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) = #1202 (order 1024) · Inn(G) = C2×C22⋊C4 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | #525 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
167 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 167 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,6,8,15)(3,10)(7,14,9,11)(12,16);(1,2)(3,11)(4,7)(5,9)(6,12)(8,13)(10,15)(14,16);(2,7)(6,14)(8,9)(11,15)Polycyclic presentation:
2^7|a^2=e,d^2=g,b^a=bd,d^a=df,c^b=cf,d^b=dg,e^b=efg32 conjugacy classes — view the full character table.
| maximal subgroup of | #8346, #8347, #8453, … |
| maximal normal subgroup of | #8346, #8347, #8453, … |
| maximal quotient of | #7578, #7579, #7586, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 236);