Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) = #72236620 (order 2048) · Inn(G) = #120 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C2×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
191 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 191 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,13)(12,16)(14,19)(15,20)(17,18);(6,10)(8,16)(11,19)(12,13)(14,17)(18,20);(2,4)(5,7,9,15)(6,11,13,20)(8,17,16,14)(10,18,12,19)Polycyclic presentation:
2^7|c^2=g,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fg,c^b=cf,d^b=dg,d^c=dg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2003, #23399, #23400, … |
| maximal normal subgroup of | C2×#2003, #23399, #23400, … |
| maximal quotient of | #9915, #9921, #9923, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 735);