Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) = C2×#72224760 (order 3072) · Inn(G) = C42 · Out(G) = C2×GL(2,ℤ/4) |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C42 |
| commutator | C4 | C2×C42 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #376 |
| prime radical | C23 | #41 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11,14,24)(4,8)(5,9)(6,10)(7,13,19,27)(12,23,26,31)(15,20)(16,21)(17,22)(18,25,29,32)(28,30);(1,3)(2,7,10,19)(4,12)(5,13)(6,14)(8,18,21,29)(9,11,22,24)(15,25)(16,26)(17,27)(20,23,30,31)(28,32);(1,4,5,15,6,16,17,28)(2,8,9,20,10,21,22,30)(3,12,13,25,14,26,27,32)(7,18,11,23,19,29,24,31)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=d,d^2=g,b^a=bd,f^a=fg,e^b=eg44 conjugacy classes — view the full character table.
| Fitting subgroup of | #1747⋊C3, #11711367, #11711368, … |
| maximal subgroup of | #11769, #11770, #11771, … |
| maximal normal subgroup of | #11769, #11770, #11771, … |
| maximal quotient of | #8137, #8138, #8139, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 479);