Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) = C22×#23900 (order 1024) · Inn(G) = #120 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C23 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,22)(3,12,14,29)(4,17,23,19)(5,10,6,11)(8,24,25,13)(9,21,32,20)(15,28,16,30)(18,26,31,27);(1,3,5,15)(2,8,10,26)(4,18,20,13)(6,16,7,14)(9,28,17,12)(11,27,22,25)(19,29,32,30)(21,24,23,31);(1,2,6,11)(3,8,16,27)(4,19,21,9)(5,10,7,22)(12,13,30,31)(14,25,15,26)(17,20,32,23)(18,29,24,28)Polycyclic presentation:
2^7|a^2=fg,b^2=f,c^2=g,d^2=f,b^a=bd,c^a=ce,d^a=df,d^b=df,e^b=eg,d^c=dg23 conjugacy classes — view the full character table.
| maximal subgroup of | #12221, #12222, #12223, … |
| maximal normal subgroup of | #12221, #12222, #12223, … |
| maximal quotient of | #7792, #7794, #7806, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 349);