Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) = C2×#6628842 (order 1024) · Inn(G) = C2×D4 · Out(G) = C2×#120 |
| 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 | |
74 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)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23);(2,6)(3,7)(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(1,3,5,6,4,7,8,2)(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=e,d^2=f,e^2=g,b^a=bd,c^a=cfg,d^a=df,c^b=cg,d^b=df32 conjugacy classes — view the full character table.
| maximal subgroup of | #12077, #12079, #12087, … |
| maximal normal subgroup of | #12077, #12079, #12087, … |
| maximal quotient of | #8209, #8287, #8850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 325);