Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) = #72231699 (order 1024) · Inn(G) = #102 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #102 |
| commutator | C8 | C22×C4 |
| Frattini, cosocle | C2×C8 | C23 |
| socle | C2 | C2×#102 |
| prime radical | C2×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,8,6,11,18,10)(3,12,15,26,16,29,13,28)(5,19,17,9,21,25,30,27)(7,20,23,14,24,31,22,32);(1,3)(2,7)(4,13)(5,20)(6,16)(8,22)(9,26)(10,23)(11,24)(12,25)(14,30)(15,18)(17,32)(19,29)(21,31)(27,28);(1,4,6,18)(2,8,11,10)(3,13,16,15)(5,17,21,30)(7,22,24,23)(9,25,27,19)(12,28,29,26)(14,20,32,31)Polycyclic presentation:
2^7|a^2=e,c^2=g,d^2=f,e^2=g,f^2=g,b^a=bd,d^a=df,f^a=fg,d^b=dfg,e^b=eg,f^b=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #12646, #12651, #12740, … |
| maximal normal subgroup of | #12646, #12651, #12740, … |
| maximal quotient of | #11597, #11598, #11651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 880);