Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C23 |
| commutator, socle | C22 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| prime radical | #532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,10,13,9,12,18,20)(7,14,16,17,15,11,19,8);(2,4)(5,7)(6,11)(8,13)(9,15)(10,16)(12,14)(17,20)(18,19);(5,8,10,14,9,17,18,11)(6,7,13,16,12,15,20,19);(6,12)(7,15)(13,20)(16,19)Polycyclic presentation:
2^7|a^2=f,c^2=f,f^2=g,b^a=be,d^a=dg,d^b=dg56 conjugacy classes — view the full character table.
| Frattini subgroup of | #94136, #94137, #94138, … |
| maximal subgroup of | #8241, #8243, #8245, … |
| maximal normal subgroup of | #8241, #8243, #8245, … |
| maximal quotient of | #17956, #17957, #17958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1663);