Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | Aut(G) = #511771 (order 512) · Inn(G) = D42 · Out(G) = D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
224 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,7,13)(3,14,18,29)(4,10,20,28)(5,12,6,11)(8,21,25,17)(9,22,26,19)(15,23,32,24)(16,30,31,27);(1,3,7,18)(2,8,13,25)(4,15,20,32)(5,21,6,17)(9,23,26,24)(10,27,28,30)(11,29,12,14)(16,22,31,19);(1,4)(2,9)(3,15)(5,19)(6,22)(7,20)(8,23)(10,11)(12,28)(13,26)(14,30)(16,21)(17,31)(18,32)(24,25)(27,29);(1,4)(2,10)(3,16)(5,22)(6,19)(7,20)(8,24)(9,11)(12,26)(13,28)(14,30)(15,21)(17,32)(18,31)(23,25)(27,29)Polycyclic presentation:
2^7|a^2=g,b^2=g,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,d^b=de,e^b=eg,f^c=fg,e^d=eg26 conjugacy classes — view the full character table.
| maximal subgroup of | #13678, #13680, C2×#3289, … |
| maximal normal subgroup of | #13678, #13680, C2×#3289, … |
| maximal quotient of | #19757, #19758, #19760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2021);