Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
286 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,13)(8,16)(9,11)(10,15)(12,17)(14,19)(18,20);(5,7,8,15,9,14,17,20)(6,10,12,13,11,18,16,19);(1,2,3,4)(6,11)(10,18)(12,16)(13,19);(7,14)(10,18)(13,19)(15,20)Polycyclic presentation:
2^7|b^2=e,c^2=fg,e^2=g,b^a=be,c^a=cf,e^a=eg,d^b=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | #9441, #10018, #10020, … |
| maximal normal subgroup of | #9441, #10018, #10020, … |
| maximal quotient of | #19623, #19624, #19627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1885);