Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) = C24×#120 (order 512) · Inn(G) = #120 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | #538 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
116 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,13,16)(11,14,18,21)(12,15,19,22)(17,23,24,20);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,17)(13,18)(15,20)(16,21)(19,24)(22,23);(1,3,5,4)(2,6,7,8)(9,12,13,19)(10,15,16,22)(11,17,18,24)(14,20,21,23)Polycyclic presentation:
2^7|a^2=g,c^2=fg,d^2=f,b^a=bd,c^a=ce,d^a=df,d^b=df,e^b=eg,d^c=dg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1612, #18121, #18126, … |
| maximal normal subgroup of | C2×#1612, #18121, #18126, … |
| maximal quotient of | #7795, #7796, #7799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 344);