Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
250 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,15)(8,14)(9,11)(10,13)(12,16)(17,20)(18,19);(1,2,3,4)(6,11)(12,19)(13,14)(15,20);(5,7)(6,12)(8,16)(9,17)(10,18)(11,19)(13,15)(14,20);(5,8,9,10)(6,13,11,14)(7,16,17,18)(12,15,19,20)Polycyclic presentation:
2^7|b^2=eg,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=dg,f^a=fg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #9349, #9350, #24473, … |
| maximal normal subgroup of | #9349, #9350, #24473, … |
| maximal quotient of | #19274, #19275, #19276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1772);