Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) = C22×#23900 (order 1024) · Inn(G) = #120 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C24 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(5,8)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18);(1,2)(3,5)(4,6)(7,8)(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(1,3)(2,5)(4,7)(6,8)Polycyclic presentation:
2^7|a^2=f,b^2=f,d^2=f,b^a=bd,c^a=ce,d^a=df,d^b=df,e^b=eg,d^c=dg23 conjugacy classes — view the full character table.
| maximal subgroup of | #12214, #12215, #12217, … |
| maximal normal subgroup of | #12214, #12215, #12217, … |
| maximal quotient of | #7756, #7759, #7764, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 334);