Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | Aut(G) = #72236189 (order 1024) · Inn(G) = #392 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #392 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | #575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
174 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 44 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)(3,13)(4,17)(5,11)(6,10)(7,12)(8,15)(9,19)(14,22)(16,27)(18,26)(20,25)(21,23)(24,30)(28,29)(31,32);(1,3)(2,8)(4,14)(5,21)(6,15)(7,16)(9,24)(10,27)(11,13)(12,23)(17,28)(18,31)(19,29)(20,30)(22,26)(25,32);(1,4,7,20)(2,9,12,26)(3,14,16,30)(5,19,6,18)(8,22,23,24)(10,17,11,25)(13,28,27,32)(15,31,21,29)Polycyclic presentation:
2^7|c^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,d^b=dg,d^c=dg,e^c=eg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2009, #23412, #23873, … |
| maximal normal subgroup of | C2×#2009, #23412, #23873, … |
| maximal quotient of | #9931, #9933, #9935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 741);