Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | Aut(G) = #72236189 (order 1024) · Inn(G) = #397 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #397 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C23 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11,7,12,5,10)(3,13,16,8,17,24,15,25)(4,18,20,26,21,27,22,9)(14,29,30,32,19,23,31,28);(1,3,7,17)(2,8,12,25)(4,19,21,14)(5,16,6,15)(9,23,26,29)(10,24,11,13)(18,32,27,28)(20,30,22,31);(1,4,5,20,7,21,6,22)(2,9,10,18,12,26,11,27)(3,14,15,30,17,19,16,31)(8,23,13,28,25,29,24,32)Polycyclic presentation:
2^7|a^2=e,b^2=g,c^2=d,d^2=g,e^2=g,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,e^c=eg,f^c=fg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2082, #23652, #24987, … |
| maximal normal subgroup of | C2×#2082, #23652, #24987, … |
| maximal quotient of | #10632, #10633, #10635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 814);