Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C24 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
304 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,3)(9,10,13,16)(11,15,12,14);(1,2,4,3)(6,8)(9,11,13,12)(10,14,16,15);(5,6)(7,8)(9,11,13,12)(10,14,16,15);(1,3,4,2)(9,12,13,11)(10,15,16,14)Polycyclic presentation:
2^7|a^2=f,b^2=e,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | #25950, #25955, #25958, … |
| maximal normal subgroup of | #25950, #25955, #25958, … |
| maximal quotient of | #13880, #13907, #13981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1172);