Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | Aut(G) = #72230894 (order 1024) · Inn(G) = #95 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #95 |
| commutator | C2×C4 | C42 |
| Frattini, cosocle | C2×C42 | C22 |
| prime radical | C23 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,17,18,14)(12,16)(13,15)(19,23,24,21)(20,22);(2,5,7,6)(9,11,12,19,13,18,20,24)(10,14,16,21,15,17,22,23)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=fg,e^2=g,b^a=bc,c^a=cf,e^a=efg,d^b=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1288, #7972, #7973, … |
| maximal normal subgroup of | C2×#1288, #7972, #7973, … |
| maximal quotient of | #7019, #7023, #7024, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 20);