Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = #72260579 (order 4096) · Inn(G) = C23 · Out(G) = C2×#10991 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C42 | C23 |
| commutator, socle | C22 | C2×C42 |
| prime radical | C23 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(6,8)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(1,2,3,6,4,5,7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^7|a^2=d,b^2=e,d^2=f,e^2=g,b^a=bg,c^b=cf56 conjugacy classes — view the full character table.
| maximal subgroup of | #7147, #7197, #7198, … |
| maximal normal subgroup of | #7147, #7197, #7198, … |
| maximal quotient of | #7843, #7849, #8133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 184);