Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1414 |
| automorphism | Aut(G) = C2×#2127 (order 256) · Inn(G) = #354 · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #354 |
| commutator | C42 | C2×C4 |
| Frattini, cosocle | #124 | C22 |
| prime radical | C2×D4 | D4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,11,13,26)(4,15,17,31)(5,19,21,9)(6,10)(7,12,22,29)(8,18)(14,28,20,27)(16,23,30,24)(25,32);(1,3)(2,7,10,24)(4,16)(5,20)(6,14)(8,11,25,28)(9,22,19,23)(12,17)(13,21)(15,26,31,27)(18,30)(29,32)Polycyclic presentation:
2^7|a^2=d,b^2=f,c^2=f,e^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,c^b=cfg,d^b=defg,e^b=eg,d^c=dfg,e^d=eg17 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1414, #13602, #13603, … |
| maximal normal subgroup of | C2×#1414, #13602, #13603, … |
| maximal quotient of | #7359, #7366, #7412, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 146);