Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | Aut(G) = #72236189 (order 1024) · Inn(G) = #394 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #394 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
127 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 127 conjugacy classes of subgroups — too many to draw. Only its 40 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:
(1,2,7,12)(3,13,16,28)(4,17,21,26)(5,11,6,10)(8,22,24,15)(9,19,27,20)(14,25,18,23)(29,32,31,30);(1,3)(2,8)(4,18)(5,22)(6,15)(7,16)(9,25)(10,28)(11,13)(12,24)(14,21)(17,31)(19,30)(20,32)(23,27)(26,29);(1,4,5,19,7,21,6,20)(2,9,10,26,12,27,11,17)(3,14,15,30,16,18,22,32)(8,23,13,29,24,25,28,31)Polycyclic presentation:
2^7|a^2=g,c^2=d,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cf,d^b=dg,f^c=fg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2049, #23525, #24295, … |
| maximal normal subgroup of | C2×#2049, #23525, #24295, … |
| maximal quotient of | #10444, #10445, #10446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 781);