Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = C2×#1202 (order 2048) · Inn(G) = C24 · Out(G) = C2×D42 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C24 |
| commutator, socle | C22 | C23×C4 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
196 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^7|a^2=f,c^2=e,f^2=g,b^a=be,d^b=dg,d^c=dg50 conjugacy classes — view the full character table.
| maximal subgroup of | #8903, #8905, #8994, … |
| maximal normal subgroup of | #8903, #8905, #8994, … |
| maximal quotient of | #20368, #20378, #20431, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1657);