Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) = #72216600 (order 1024) · Inn(G) = D42 · Out(G) = C2×D4 |
| 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 | |
230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,7,12)(3,13,14,15)(4,9,8,16)(5,11,6,10);(1,3)(2,8)(4,12)(5,16)(6,9)(7,14)(10,15)(11,13);(1,4)(2,3)(5,15)(6,13)(7,8)(9,10)(11,16)(12,14);(1,4)(2,9)(3,10)(5,13)(6,15)(7,8)(11,14)(12,16)Polycyclic presentation:
2^7|a^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 | #12314, #12315, #12364, … |
| maximal normal subgroup of | #12314, #12315, #12364, … |
| maximal quotient of | #19884, #19888, #19890, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2023);