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) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #99 |
| commutator | C22×C4 | C2×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C22 |
| prime radical | C23 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,5,6,7)(9,10,12,16,13,15,19,22)(11,14,18,20,17,21,24,23);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14,15,21)(12,18)(13,17)(16,23,22,20)(19,24)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=fg,d^2=g,b^a=bc,c^a=ce,e^a=eg,c^b=cf,d^b=defg,d^c=dg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1353, #11721, #11722, … |
| maximal normal subgroup of | C2×#1353, #11721, #11722, … |
| maximal quotient of | #7122, #7132, #7251, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 85);