Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | Aut(G) = #72240903 (order 6144) · Inn(G) = C2×#120 · Out(G) = C22×S4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#120 |
| commutator, Frattini, cosocle | C23 | C24 |
| prime radical | #585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,7,13,20)(4,8,15,22)(5,16,17,9)(11,23,25,18)(12,26,27,19)(14,28,29,21)(24,30,32,31);(1,3)(2,7)(4,11)(5,12)(6,13)(8,18)(9,19)(10,20)(14,24)(15,25)(16,26)(17,27)(21,30)(22,23)(28,31)(29,32);(1,4)(2,8)(3,11)(5,14)(6,15)(7,18)(9,21)(10,22)(12,24)(13,25)(16,28)(17,29)(19,30)(20,23)(26,31)(27,32);(1,5,6,17)(2,9,10,16)(3,12,13,27)(4,14,15,29)(7,19,20,26)(8,21,22,28)(11,24,25,32)(18,30,23,31)Polycyclic presentation:
2^7|a^2=g,d^2=g,b^a=be,c^a=cf,d^a=dg,f^b=fg,e^c=eg29 conjugacy classes — view the full character table.
| Fitting subgroup of | #3028⋊C3, #11711639, #11711641, … |
| maximal subgroup of | #13351, #13352, #13354, … |
| maximal normal subgroup of | #13351, #13352, #13354, … |
| maximal quotient of | #18308, #18309, #18310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1760);