Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
125 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 125 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,9,11)(7,12,8,10)(13,14)(15,18)(16,19)(17,20);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,17,16)(14,18,20,19);(2,4)(5,8,9,7)(6,10,11,12)(14,19)(15,16)(18,20)Polycyclic presentation:
2^7|a^2=d,b^2=dg,c^2=d,e^2=g,b^a=bd,c^a=ce,e^a=eg,c^b=cf,e^c=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2059, #23551, #23553, … |
| maximal normal subgroup of | C2×#2059, #23551, #23553, … |
| maximal quotient of | #10571, #10572, #10576, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 791);