Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1366 |
| automorphism | Aut(G) = #6354100 (order 512) · Inn(G) = #333 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #333 |
| commutator | C42 | C2×C4 |
| Frattini, cosocle | C4×C8 | C22 |
| prime radical | C22 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
37 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,17,10,28,26,6,11,19,27,5,8,18,24)(3,12,14,25,20,23,16,31,13,29,30,32,15,7,21,22);(1,3,6,13)(2,7)(4,16,19,21)(5,20,17,15)(8,12)(9,25)(10,29)(11,23)(14,18,30,28)(22,26)(24,31)(27,32)Polycyclic presentation:
2^7|a^2=d,b^2=f,c^2=fg,d^2=ef,e^2=g,b^a=bc,c^a=ce,e^a=eg,f^a=fg,c^b=cf,d^b=def,e^b=eg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1366, #11956, #11989, … |
| maximal normal subgroup of | C2×#1366, #11956, #11989, … |
| maximal quotient of | #7135, #7271, C2×#1366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 98);