Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#575 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C25 |
| socle | C22 | C22×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
742 conjugacy classes of subgroups; only the 428 normal subgroups are shown.
This group has 742 conjugacy classes of subgroups — too many to draw. Only its 428 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4)(5,12)(6,11)(7,15)(8,10)(9,13)(14,18)(16,17);(3,5)(4,9)(6,13)(7,16)(8,14)(10,17)(11,18)(12,15);(4,10)(9,17)(11,15)(12,18);(3,6,8,7)(4,11,10,15)(5,13,14,16)(9,18,17,12)44 conjugacy classes — view the full character table.
| maximal subgroup of | #11018, #11019, #11032, … |
| maximal normal subgroup of | #11018, #11019, #11032, … |
| maximal quotient of | C2×#2896, C2×#2911, C2×#2943, … |
| automorphism group of | #364 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2312);