Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) = #72236881 (order 1024) · Inn(G) = #121 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| socle | C2 | C2×#121 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
226 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 98 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)(3,13)(4,18)(5,10)(6,11)(7,12)(8,15)(9,19)(14,28)(16,24)(17,23)(20,25)(21,26)(22,30)(27,29)(31,32);(1,3,5,15,7,17,6,16)(2,8,11,13,12,24,10,23)(4,14,19,29,20,30,21,31)(9,22,18,27,26,28,25,32);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,25)(11,18)(12,26)(13,27)(15,29)(16,31)(17,30)(23,32)(24,28);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,20,21)(8,23,24,13)(9,25,26,18)(14,29,30,31)(22,32,28,27)Polycyclic presentation:
2^7|b^2=e,d^2=g,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #12995, #13005, #13070, … |
| maximal normal subgroup of | #12995, #13005, #13070, … |
| maximal quotient of | #19101, #19105, #19110, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1799);