Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C23 |
| commutator | C23 | C22×C4 |
| prime radical | C25 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(6,8)(9,10,11,12)(13,14,16,15);(2,4)(6,8)(14,15);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15,16,14)Polycyclic presentation:
2^7|a^2=g,c^2=d,b^a=bd,c^a=ce,c^b=cf44 conjugacy classes — view the full character table.
| maximal subgroup of | #7324, #8535, C2×#1440, … |
| maximal normal subgroup of | #7324, #8535, C2×#1440, … |
| maximal quotient of | #7555, #7556, #11147, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 172);