Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | Aut(G) = #72247039 (order 6144) · Inn(G) = C23×D4 · Out(G) = C22×S4 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C23×D4 |
| commutator, Frattini, cosocle | C4 | C25 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
732 conjugacy classes of subgroups; only the 426 normal subgroups are shown.
This group has 732 conjugacy classes of subgroups — too many to draw. Only its 426 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,7)(4,6)(5,8)(9,12)(10,14)(11,13)(15,16);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15);(1,3)(2,7)(4,11)(5,10)(6,13)(8,14)(9,15)(12,16);(3,10)(7,14)(11,16)(13,15);(2,8)(6,9)(7,14)(13,15)Polycyclic presentation:
2^7|f^2=g,b^a=bf,e^a=eg,f^a=fg,f^b=fg,d^c=dg41 conjugacy classes — view the full character table.
| Fitting subgroup of | #3585⋊C3, #11713196, #11713199, … |
| maximal subgroup of | #23872, #23874, #23882, … |
| maximal normal subgroup of | #23872, #23874, #23882, … |
| maximal quotient of | #60087, #60089, #60109, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2317);