Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | Aut(G) = D4×#442 (order 512) · Inn(G) = #447 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,14)(4,18)(5,12)(6,11)(7,13)(8,15)(9,16)(10,21)(17,29)(19,30)(20,24)(22,28)(23,27)(25,32)(26,31);(1,3)(2,8)(4,19)(5,16)(6,23)(7,17)(9,24)(10,27)(11,25)(12,30)(13,26)(14,18)(15,21)(20,31)(22,32)(28,29);(1,3)(2,9)(4,15)(5,23)(6,16)(7,17)(8,24)(10,26)(11,29)(12,14)(13,27)(18,30)(19,21)(20,32)(22,31)(25,28)Polycyclic presentation:
2^7|d^2=fg,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fg,d^b=df,f^b=fg,e^c=eg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2192, #33538, #33539, … |
| maximal normal subgroup of | C2×#2192, #33538, #33539, … |
| maximal quotient of | #12108, #12110, #12164, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 924);