Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) = C2×#63011 (order 512) · Inn(G) = #120 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #120 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #386 |
| prime radical | #575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
155 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 155 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12,15,23)(4,16)(5,10)(6,20)(7,11)(8,14,24,19)(9,17)(13,28,31,22)(18,25)(21,26)(27,29,32,30);(1,3,5,19,7,15,6,14)(2,8,10,23,11,24,20,12)(4,13,17,30,18,31,21,29)(9,22,25,32,26,28,16,27);(1,4)(2,9)(3,13)(5,17)(6,21)(7,18)(8,22)(10,25)(11,26)(12,27)(14,29)(15,31)(16,20)(19,30)(23,32)(24,28)Polycyclic presentation:
2^7|a^2=f,b^2=df,d^2=g,e^2=g,b^a=bd,c^a=ce,e^a=eg,d^b=dg,f^b=fg,e^c=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1881, #20939, #21477, … |
| maximal normal subgroup of | C2×#1881, #20939, #21477, … |
| maximal quotient of | #8953, #8954, #8955, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 613);