Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | #121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
164 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,27)(20,26)(21,23)(22,25)(24,28)(29,32)(30,31);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(18,23)(24,31)(25,26)(27,32);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,21,29)(18,24,23,31)(20,28,22,30)(25,27,26,32);(2,8)(6,9)(7,16)(14,15)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,27,31,32)Polycyclic presentation:
2^7|b^2=eg,c^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ceg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | #9369, #19011, #19013, … |
| maximal normal subgroup of | #9369, #19011, #19013, … |
| maximal quotient of | #21842, #21868, #21989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1972);