Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2032 |
| automorphism | Aut(G) = #72216883 (order 1536) · Inn(G) = #393 · Out(G) = C2×A4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #393 |
| commutator, Frattini, cosocle | C24 | C23 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
139 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,6)(4,11,13,7)(5,8)(9,16,12,14)(10,15);(1,3)(2,6)(4,12,13,9)(5,10)(7,14,11,16)(8,15);(1,4)(2,7)(3,9)(5,13)(6,14)(8,11)(10,12)(15,16)Polycyclic presentation:
2^7|a^2=d,b^2=d,b^a=bd,c^a=ce,e^a=eg,f^a=fg,c^b=cf,f^b=fg,d^c=dg23 conjugacy classes — view the full character table.
| Fitting subgroup of | #2032⋊C3 |
| maximal subgroup of | C2×#2032, #23474, #23996, … |
| maximal normal subgroup of | C2×#2032, #23474, #23996, … |
| maximal quotient of | #10374, #10375, #10376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 764);