Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) = C2×#511819 (order 1024) · Inn(G) = #127 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #399 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
105 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,11,7,21)(3,12,14,24,16,28,15,8)(4,17,19,27,20,9,22,26)(13,29,18,32,30,25,31,23);(1,3,7,14,6,16,5,15)(2,8,21,12,11,24,10,28)(4,18,22,30,20,31,19,13)(9,25,27,23,17,29,26,32);(1,4)(2,9)(3,13)(5,19)(6,20)(7,22)(8,23)(10,26)(11,17)(12,29)(14,18)(15,31)(16,30)(21,27)(24,32)(25,28)Polycyclic presentation:
2^7|a^2=d,b^2=de,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=fg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7438, #7439, #12301, … |
| maximal normal subgroup of | #7438, #7439, #12301, … |
| maximal quotient of | #10659, #10660, #10661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 830);