Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) = #72256975 (order 4096) · Inn(G) = C2×D8 · Out(G) = C2×#525 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | C2×D16 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,15)(8,20)(9,12)(10,23)(11,17)(13,16)(14,21)(18,27)(19,34)(22,35)(24,33)(25,36)(26,31)(28,30)(29,32);(5,7,16,30,10,19,22,33,9,18,31,36,21,32,8,17)(6,11,20,29,14,25,26,27,12,24,35,34,23,28,13,15);(1,2,3,4)(6,12)(11,24)(13,26)(14,23)(15,27)(20,35)(25,28)(29,34)Polycyclic presentation:
2^7|b^2=df,c^2=eg,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 | #7446, #7447, #13689, … |
| maximal normal subgroup of | #7446, #7447, #13689, … |
| maximal quotient of | #12503, #12504, #12505, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 981);