Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#10586169 (order 1024) · Inn(G) = C22 · Out(G) = C22×#585 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C22 |
| commutator | C2 | C22×C16 |
| Frattini, cosocle | C2×C8 | C23 |
| socle | C22 | C22×C8 |
| prime radical | C23 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(2,5)(6,8);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^7|a^2=e,c^2=d,e^2=f,f^2=g,b^a=bd80 conjugacy classes — view the full character table.
| maximal subgroup of | #7142, #7144, #7145, … |
| maximal normal subgroup of | #7142, #7144, #7145, … |
| maximal quotient of | #12590, C4×C22⋊C16, C4×C4⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 894);