Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) = #72238442 (order 8192) · Inn(G) = C2×C22⋊C4 · Out(G) = #23900 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C23 | C22×C4 |
| Frattini, prime radical, cosocle | C24 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22);(2,5)(6,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)Polycyclic presentation:
2^7|a^2=f,b^2=g,c^2=d,b^a=bd,c^a=ce,e^a=eg,f^c=fg32 conjugacy classes — view the full character table.
| Frattini subgroup of | #94526, #94527, #94550, … |
| maximal subgroup of | #7214, #7348, #8463, … |
| maximal normal subgroup of | #7214, #7348, #8463, … |
| maximal quotient of | #8699, #8701, #8703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 552);