Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) = C24×#442 (order 1024) · Inn(G) = C2×D8 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | C2×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
79 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,15)(8,11)(9,21)(10,24)(12,17)(13,23)(14,20)(16,31)(18,35)(19,32)(22,30)(25,29)(26,33)(27,34)(28,36);(1,2,3,4)(6,11)(12,26)(13,27)(14,24)(15,31)(21,32)(28,30)(29,35);(5,7,8,16)(6,12,11,26)(9,22,19,36)(10,25,20,18)(13,29,27,35)(14,30,24,28)(15,32,31,21)(17,34,33,23)Polycyclic presentation:
2^7|b^2=dg,c^2=g,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,e^c=ef,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2211, #33579, #33581, … |
| maximal normal subgroup of | C2×#2211, #33579, #33581, … |
| maximal quotient of | #12419, #12423, #12429, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 943);