Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C2×#72215858 (order 2048) · Inn(G) = C23 · Out(G) = C23×#120 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C23 |
| commutator, socle | C22 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| prime radical | #525 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 144 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)(5,6,8,12,9,11,15,18)(7,10,13,17,14,16,19,20);(2,4);(5,7,8,13,9,14,15,19)(6,10,12,17,11,16,18,20);(6,11)(10,16)(12,18)(17,20)Polycyclic presentation:
2^7|a^2=f,c^2=f,f^2=g,b^a=be,d^a=dg56 conjugacy classes — view the full character table.
| Frattini subgroup of | #94381, #94382, #94397, … |
| maximal subgroup of | #8238, #8239, #8240, … |
| maximal normal subgroup of | #8238, #8239, #8240, … |
| maximal quotient of | #17900, #17901, #17902, … |
| outer automorphism group of | C2×D32, C2×Q64 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1661);