Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) = C24×D42 (order 1024) · Inn(G) = #120 · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | #535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
163 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 163 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,3)(5,6)(7,9)(8,11)(10,12);(1,2)(3,4)(7,12)(9,10);(1,3,4,2)(6,9)(7,12)(10,11)Polycyclic presentation:
2^7|c^2=d,e^2=g,b^a=bd,c^a=ce,e^a=eg,f^a=fg,c^b=cf,e^b=eg,e^c=eg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2037, #23492, #23493, … |
| maximal normal subgroup of | C2×#2037, #23492, #23493, … |
| maximal quotient of | #10394, #10395, #10398, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 769);