Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | Aut(G) = #72236189 (order 1024) · Inn(G) = C2×#121 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#121 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | #575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
223 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 223 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 4 generators on 32 points:
(1,2,7,12)(3,13,15,24)(4,18,20,26)(5,11,6,10)(8,17,23,16)(9,21,25,19)(14,28,29,22)(27,31,32,30);(1,3,5,16,7,15,6,17)(2,8,10,13,12,23,11,24)(4,14,19,30,20,29,21,31)(9,22,18,27,25,28,26,32);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,18)(11,26)(12,25)(13,27)(15,29)(16,30)(17,31)(23,28)(24,32);(3,15)(8,23)(13,24)(14,29)(16,17)(22,28)(27,32)(30,31)Polycyclic presentation:
2^7|a^2=g,b^2=e,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,d^b=dg,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | #24421, #24423, #24474, … |
| maximal normal subgroup of | #24421, #24423, #24474, … |
| maximal quotient of | #19163, #19164, #19167, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1801);