Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) = C2×#7622880 (order 1024) · Inn(G) = #120 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #120 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #379 |
| prime radical | #585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
274 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 3 generators on 32 points:
(1,2,6,10,7,11,5,18)(3,12,14,8,15,23,19,22)(4,9,16,25,17,26,20,24)(13,28,31,32,30,27,29,21);(1,3)(2,8)(4,13)(5,19)(6,14)(7,15)(9,21)(10,23)(11,22)(12,18)(16,31)(17,30)(20,29)(24,27)(25,28)(26,32);(1,4)(2,9)(3,13)(5,20)(6,16)(7,17)(8,21)(10,25)(11,26)(12,27)(14,31)(15,30)(18,24)(19,29)(22,32)(23,28)Polycyclic presentation:
2^7|a^2=f,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dg,e^b=eg,d^c=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7346, #7347, #13045, … |
| maximal normal subgroup of | #7346, #7347, #13045, … |
| maximal quotient of | #8318, #8319, #8320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 526);