Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
550 conjugacy classes of subgroups; only the 430 normal subgroups are shown.
This group has 550 conjugacy classes of subgroups — too many to draw. Only its 430 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) = 32 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,15,16,13);(2,5)(6,7);(9,11,12,16)(10,13,14,15);(1,2)(3,6)(4,5)(7,8);(1,3)(2,6)(4,8)(5,7)Polycyclic presentation:
2^7|a^2=g,c^2=g,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| maximal subgroup of | #8351, #8355, #8432, … |
| maximal normal subgroup of | #8351, #8355, #8432, … |
| maximal quotient of | #34181, #34212, #34213, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2233);