Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) = C2×#511819 (order 1024) · Inn(G) = #124 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #400 |
| prime radical | C23 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,11,7,22)(3,12,14,24,16,27,15,8)(4,17,19,31,20,9,21,26)(13,25,28,32,29,30,18,23);(1,3,7,14,6,16,5,15)(2,8,22,12,11,24,10,27)(4,18,21,13,20,28,19,29)(9,25,31,32,17,30,26,23);(1,4,5,19,6,20,7,21)(2,9,10,26,11,17,22,31)(3,13,14,28,16,29,15,18)(8,23,12,25,24,32,27,30)Polycyclic presentation:
2^7|a^2=d,b^2=de,c^2=d,d^2=g,f^2=g,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | #12304, #12305, #12306, … |
| maximal normal subgroup of | #12304, #12305, #12306, … |
| maximal quotient of | #10687, #10688, #10689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 835);