Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C22×#11709839 (order 3072) · Inn(G) = C23 · Out(G) = C2×#392⋊C3 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = 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 | |
89 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8,5,7,11,14)(3,6,10,12,9,13,15,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,28)(27,30)(31,32);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,20,25,21,26,27,31)(18,22,24,29,23,28,30,32);(2,7)(3,9)(8,14)(10,15)(18,23)(22,28)(24,30)(29,32)Polycyclic presentation:
2^7|a^2=d,b^2=e,d^2=f,e^2=g,c^a=cfg,c^b=cf56 conjugacy classes — view the full character table.
| Fitting subgroup of | #1454⋊C3 |
| maximal subgroup of | C2×#1454, #17324, #17326, … |
| maximal normal subgroup of | C2×#1454, #17324, #17326, … |
| maximal quotient of | #7845, #7851, #8135, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 186);