Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) = #72240975 (order 2048) · Inn(G) = D42 · Out(G) = #120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,14)(4,10)(5,12)(6,11)(7,13)(8,15)(9,16);(1,3,7,9)(2,8,13,4)(5,10,6,15)(11,16,12,14);(1,4)(2,9)(3,13)(5,16)(6,14)(7,8)(10,11)(12,15);(1,4)(2,10)(3,12)(5,14)(6,16)(7,8)(9,11)(13,15)Polycyclic presentation:
2^7|b^2=g,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,c^b=cg,d^b=de,e^b=eg,f^c=fg,e^d=eg29 conjugacy classes — view the full character table.
| maximal subgroup of | #12310, #12312, #12366, … |
| maximal normal subgroup of | #12310, #12312, #12366, … |
| maximal quotient of | #19885, #19893, #19898, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2024);