Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) = #72256567 (order 2048) · Inn(G) = C24 · Out(G) = C2×#534 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C24 |
| commutator, socle | C22 | C23×C4 |
| prime radical | #532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,17,12,16,20,24)(11,15,19,23,18,14,21,22);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,16)(17,22)(20,21)(23,24);(3,8)(5,7)(9,11)(10,15)(12,18)(13,19)(14,16)(17,23)(20,21)(22,24);(2,6)(5,7)(10,16)(11,18)(17,24)(19,21)Polycyclic presentation:
2^7|a^2=f,f^2=g,b^a=be,d^a=de,c^b=ceg,d^b=dg,d^c=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #9279, #9281, #9283, … |
| maximal normal subgroup of | #9279, #9281, #9283, … |
| maximal quotient of | #20531, #20537, #20565, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1713);