Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | Aut(G) = C23×B(4,𝔽2) (order 512) · Inn(G) = #99 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #99 |
| commutator | C22×C4 | C2×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C22 |
| prime radical | C24 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
73 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,17)(12,20,21,15)(13,16)(14,18,22,24)(19,23);(2,5)(3,8)(6,7)(9,11,13,19)(10,14,16,22)(12,18,21,24)(15,17,20,23)Polycyclic presentation:
2^7|a^2=d,b^2=g,c^2=f,b^a=bc,c^a=ce,e^a=eg,c^b=cf,d^b=def,d^c=dg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1344, #11691, #11692, … |
| maximal normal subgroup of | C2×#1344, #11691, #11692, … |
| maximal quotient of | #7077, #7079, #7085, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 76);