Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) = #72256975 (order 4096) · Inn(G) = D8 · Out(G) = #34655 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D8 |
| commutator | C8 | C22×C4 |
| Frattini, cosocle | C2×C8 | C23 |
| socle | C22 | C2×D8 |
| prime radical | C23 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
76 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)(7,16)(8,13)(9,23)(10,25)(11,15)(12,19)(14,17)(18,29)(20,35)(21,31)(22,28)(24,36)(26,33)(27,34)(30,32);(5,7,17,32,10,20,24,34,11,21,22,33,8,18,9,19)(6,12,23,29,13,26,28,31,15,27,36,35,25,30,14,16);(5,8,11,10)(6,13,15,25)(7,18,21,20)(9,22,24,17)(12,26,27,30)(14,23,28,36)(16,29,31,35)(19,33,34,32)Polycyclic presentation:
2^7|a^2=e,b^2=df,c^2=g,d^2=fg,f^2=g,b^a=bd,d^a=df,f^a=fg44 conjugacy classes — view the full character table.
| maximal subgroup of | #7473, #7475, #7477, … |
| maximal normal subgroup of | #7473, #7475, #7477, … |
| maximal quotient of | #12008, #12024, #12034, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 890);