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 |
| prime radical | #548 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
236 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2)(3,4)(6,10)(8,16)(11,19)(12,17)(13,20)(14,18);(5,6,9,12)(7,13,15,19)(8,17,16,10)(11,18,20,14);(2,4);(5,7,9,15)(6,11,12,20)(8,18,16,14)(10,19,17,13)Polycyclic presentation:
2^7|b^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,d^b=de,e^b=eg,e^d=eg29 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3270, #61030, #61053, … |
| maximal normal subgroup of | C2×#3270, #61030, #61053, … |
| maximal quotient of | #19761, #19769, #19781, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2002);