Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×#72214076 (order 2048) · Inn(G) = D4 · Out(G) = C2×#2403 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | D4 |
| commutator | C4 | C2×C42 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C22 | C4×D4 |
| prime radical | C23 | #41 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(6,10,12,15)(11,17,18,19);(1,2,3,4)(5,6)(7,11)(8,15)(9,12)(10,16)(13,19)(14,18)(17,20);(5,7,8,13,9,14,16,20)(6,11,15,19,12,18,10,17)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=de,d^2=g,b^a=bd,d^b=dg,e^b=eg56 conjugacy classes — view the full character table.
| maximal subgroup of | #7179, #7180, #7181, … |
| maximal normal subgroup of | #7179, #7180, #7181, … |
| maximal quotient of | #8266, #8267, #8270, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 504);