Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) = C23×#3003 (order 1024) · Inn(G) = C2×D4 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C42 | C23 |
| socle, prime radical | C22 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,3,4,6,8,7)(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18);(2,6)(3,7)(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17);(1,3,5,6,4,7,8,2)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=ef,d^2=f,e^2=g,b^a=bd,c^a=cg,d^a=df,c^b=cg,d^b=df44 conjugacy classes — view the full character table.
| maximal subgroup of | #7073, #7075, #12070, … |
| maximal normal subgroup of | #7073, #7075, #12070, … |
| maximal quotient of | #8207, #8282, #8848, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 323);