Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) = #72233213 (order 4096) · Inn(G) = C2×C22⋊C4 · Out(G) = C2×B(4,𝔽2) |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C42 | C23 |
| prime radical | C22×D4 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,17,19,24)(12,16)(13,15)(14,18,22,20)(21,23);(2,5)(3,7)(9,11)(10,14)(12,20)(13,19)(15,22)(16,24)(17,23)(18,21);(10,15)(14,22)(16,23)(17,24)Polycyclic presentation:
2^7|a^2=e,d^2=g,e^2=f,b^a=bd,c^a=cg,d^a=df,d^b=dg,e^b=efg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7481, #7495, #8522, … |
| maximal normal subgroup of | #7481, #7495, #8522, … |
| maximal quotient of | #7592, #7595, #7601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 265);