Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C2×#10586169 (order 1024) · Inn(G) = C23 · Out(G) = C27 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C23 |
| commutator, socle | C22 | C22×C8 |
| prime radical | C23 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
76 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)(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,14)(15,20)(16,21)(22,24)Polycyclic presentation:
2^7|a^2=e,c^2=d,e^2=f,f^2=g,b^a=bd,c^a=cg56 conjugacy classes — view the full character table.
| maximal subgroup of | #7143, #7144, #7146, … |
| maximal normal subgroup of | #7143, #7144, #7146, … |
| maximal quotient of | #12591, #12599, #12653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 895);