Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) = C2×#511819 (order 1024) · Inn(G) = #124 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #394 |
| prime radical | #575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12)(4,17)(5,22)(6,10)(7,11)(8,14)(9,19)(13,23)(15,25)(16,24)(18,26)(20,27)(21,30)(28,31)(29,32);(1,3)(2,8)(4,18)(5,15)(6,14)(7,16)(9,26)(10,24)(11,25)(12,22)(13,20)(17,23)(19,29)(21,28)(27,31)(30,32);(1,4,5,19,7,21,6,20)(2,9,10,17,11,27,22,30)(3,13,14,28,16,29,15,18)(8,23,12,26,25,32,24,31)Polycyclic presentation:
2^7|c^2=d,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cf,d^b=dg,f^b=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | #12170, #12171, #12172, … |
| maximal normal subgroup of | #12170, #12171, #12172, … |
| maximal quotient of | #10464, #10465, #10466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 785);