Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 1024 |
| 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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
249 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 249 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(1,2)(3,4)(5,6,9,12)(7,13,14,20)(8,17,18,11)(10,15,19,16);(5,7,9,14)(6,10,12,19)(8,15,18,16)(11,20,17,13);(2,4)(7,14)(10,19)(13,20)(15,16);(7,15)(8,18)(10,20)(11,17)(13,19)(14,16)Polycyclic presentation:
2^7|a^2=g,b^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,c^b=cg,d^b=de,e^b=eg,e^d=eg29 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3275, #61038, #61059, … |
| maximal normal subgroup of | C2×#3275, #61038, #61059, … |
| maximal quotient of | #19886, #19893, #19894, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2007);