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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#120 |
| commutator, Frattini, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(4,11)(6,12)(7,14)(9,15);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)Polycyclic presentation:
2^7|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 | #3027⋊C3, #11711638, #11711640, … |
| maximal subgroup of | #13350, #13351, #13353, … |
| maximal normal subgroup of | #13350, #13351, #13353, … |
| maximal quotient of | #18300, #18301, #18302, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1759);