Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) = #72240975 (order 2048) · Inn(G) = D42 · Out(G) = #120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,13)(3,14,17,19)(4,16,9,23)(5,10,22,28)(6,12,7,11)(15,30,32,26)(18,31,25,20)(21,29,24,27);(1,3,8,17)(2,9,13,4)(5,20,22,31)(6,23,7,16)(10,26,28,30)(11,19,12,14)(15,24,32,21)(18,27,25,29);(1,4,8,9)(2,3,13,17)(5,18,22,25)(6,19,7,14)(10,15,28,32)(11,16,12,23)(20,29,31,27)(21,30,24,26);(1,5)(2,10)(3,15)(4,18)(6,24)(7,21)(8,22)(9,25)(11,29)(12,27)(13,28)(14,30)(16,31)(17,32)(19,26)(20,23)Polycyclic presentation:
2^7|a^2=g,b^2=g,c^2=g,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,c^b=cg,d^b=de,e^b=eg,f^c=fg,e^d=eg29 conjugacy classes — view the full character table.
| maximal subgroup of | #12311, #12313, #12367, … |
| maximal normal subgroup of | #12311, #12313, #12367, … |
| maximal quotient of | #19887, #19894, #19902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2025);