Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) = #72231760 (order 1024) · Inn(G) = C22⋊C4 · Out(G) = C2×#121 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22⋊C4 |
| commutator | C2×C4 | C42 |
| Frattini, cosocle | C2×C42 | C22 |
| socle | C22 | #95 |
| prime radical | C23 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,12,16,14,17,13,22)(11,18,20,24,19,15,21,23);(2,5,7,6)(9,11,13,21,14,19,12,20)(10,15,16,23,17,18,22,24)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=f,d^2=g,e^2=g,b^a=bc,c^a=cf,e^a=efg,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1302, #8084, #8085, … |
| maximal normal subgroup of | C2×#1302, #8084, #8085, … |
| maximal quotient of | #7027, #7029, #7030, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 34);