Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | Aut(G) = #72236721 (order 2048) · Inn(G) = C2×#120 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#120 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | #585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
352 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 352 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12)(3,13,16,25)(4,9,18,24)(5,11,6,10)(8,19,22,15)(14,28,31,27)(17,26,20,23)(21,29,32,30);(1,3)(2,8)(4,14)(5,15)(6,19)(7,16)(9,21)(10,13)(11,25)(12,22)(17,29)(18,31)(20,30)(23,27)(24,32)(26,28);(1,4)(2,9)(3,14)(5,17)(6,20)(7,18)(8,21)(10,23)(11,26)(12,24)(13,27)(15,29)(16,31)(19,30)(22,32)(25,28);(1,5,7,6)(2,10,12,11)(3,15,16,19)(4,17,18,20)(8,13,22,25)(9,23,24,26)(14,29,31,30)(21,27,32,28)Polycyclic presentation:
2^7|a^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fg,e^c=eg29 conjugacy classes — view the full character table.
| maximal subgroup of | #13355, #13356, #13361, … |
| maximal normal subgroup of | #13355, #13356, #13361, … |
| maximal quotient of | #18136, #18140, #18146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1754);