Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | Aut(G) = #72230859 (order 1024) · Inn(G) = #95 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #95 |
| commutator | C2×C4 | C42 |
| Frattini, cosocle | C2×C42 | C22 |
| prime radical | C23 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
63 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,6,10,5,8)(3,11,12,7,14,15,13,16)(17,18)(19,25,27,31)(20,23)(21,24)(22,26,29,32)(28,30);(1,3,5,13,6,14,4,12)(2,7,9,15,10,16,8,11)(17,19)(18,22,24,29)(20,26)(21,27)(23,25,30,31)(28,32)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=fg,d^2=f,e^2=f,b^a=bc,c^a=cf,e^a=efg,d^b=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1287, #7964, #7965, … |
| maximal normal subgroup of | C2×#1287, #7964, #7965, … |
| maximal quotient of | #7018, #7021, #7023, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 19);