Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C2×#121 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
426 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 124 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:
(6,8)(9,10,13,16)(11,15,12,14);(2,4)(6,8)(9,11,13,12)(10,14,16,15);(1,2)(3,4)(5,6,7,8)(9,10)(11,14)(12,15)(13,16);(9,12,13,11)(10,15,16,14)Polycyclic presentation:
2^7|a^2=e,b^2=e,c^2=ef,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 | #9466, #9948, #9956, … |
| maximal normal subgroup of | #9466, #9948, #9956, … |
| maximal quotient of | #13881, #13922, #13978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1168);