Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C2×#20579 (order 512) · Inn(G) = C23 · Out(G) = C22×M16 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C23 |
| commutator, socle | C22 | C22×C8 |
| prime radical | C23 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
69 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,16,13,17,22,28,14,18,23,29,24,30,34,38)(11,15,20,26,21,27,33,37,19,25,31,35,32,36,39,40);(2,5)(6,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,26)(17,27)(18,25)(22,33)(23,31)(24,32)(28,37)(29,35)(30,36)(34,39)(38,40);(1,3,4,7)(2,6,5,8)(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40)Polycyclic presentation:
2^7|a^2=e,c^2=d,e^2=f,f^2=g,b^a=bd,c^b=cg56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2164, #33419, #33420, … |
| maximal normal subgroup of | C2×#2164, #33419, #33420, … |
| maximal quotient of | #12592, #12600, #12654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 896);