Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) = C22×#495725 (order 2048) · Inn(G) = C2×D8 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | C23 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
77 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,14)(7,15,19,31)(8,20,22,35)(9,23,21,13)(11,24,26,17)(12,27,28,16)(18,34,33,30)(25,32,29,36);(5,7,16,32,9,18,22,24,10,19,27,36,21,33,8,17)(6,11,20,30,13,25,28,31,14,26,35,34,23,29,12,15);(2,4)Polycyclic presentation:
2^7|a^2=g,b^2=df,d^2=fg,f^2=g,b^a=bd,c^a=ce,d^a=df,f^a=fg38 conjugacy classes — view the full character table.
| maximal subgroup of | #7419, #7421, #13664, … |
| maximal normal subgroup of | #7419, #7421, #13664, … |
| maximal quotient of | #12431, #12440, #12452, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 948);