Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#6454041 (order 1024) · Inn(G) = C22 · Out(G) = C2×D4×#41 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C22 |
| commutator | C2 | C2×C4×C8 |
| Frattini, cosocle | C2×C8 | C23 |
| socle | C22 | C22×C8 |
| prime radical | C23 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,8,12,9,13,18,24,10,14,19,25,20,26,30,34)(7,11,16,22,17,23,29,33,15,21,27,31,28,32,35,36);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,15)(12,22)(13,23)(14,21)(18,29)(19,27)(20,28)(24,33)(25,31)(26,32)(30,35)(34,36);(7,15)(11,21)(16,27)(17,28)(22,31)(23,32)(29,35)(33,36)Polycyclic presentation:
2^7|a^2=d,b^2=e,d^2=f,f^2=g,c^b=cg80 conjugacy classes — view the full character table.
| Frattini subgroup of | #94003, #94004, #94210, … |
| maximal subgroup of | #7469, #7470, #12026, … |
| maximal normal subgroup of | #7469, #7470, #12026, … |
| maximal quotient of | #11399, #11401, C16×M16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 840);