Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
236 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,10,14)(7,11,16,20)(8,13,9,18)(12,17,19,15);(1,2,3,4)(5,7)(6,11)(8,17)(9,15)(10,16)(12,18)(13,19)(14,20);(5,7)(6,12)(8,15)(9,17)(10,16)(11,18)(13,20)(14,19);(5,8,10,9)(6,13,14,18)(7,15,16,17)(11,12,20,19)Polycyclic presentation:
2^7|a^2=g,b^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=dg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #24458, #24459, #24462, … |
| maximal normal subgroup of | #24458, #24459, #24462, … |
| maximal quotient of | #19414, #19415, #19416, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1778);