Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | Aut(G) = C2×#72233929 (order 4096) · Inn(G) = #127 · Out(G) = C2×#392 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #127 |
| commutator, Frattini, cosocle | C42 | C23 |
| prime radical | #127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
93 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,6,3)(4,5,8,7)(9,10,13,16)(11,17,19,24)(12,21,18,15)(14,22,23,20);(1,3,5,8,6,2,7,4)(9,11,18,22,13,19,12,20)(10,14,21,24,16,23,15,17);(1,4,5,3,6,8,7,2)(11,19)(14,23)(17,24)(20,22)Polycyclic presentation:
2^7|a^2=f,b^2=dfg,c^2=e,d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,c^b=cf26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1720, #19613, #19619, … |
| maximal normal subgroup of | C2×#1720, #19613, #19619, … |
| maximal quotient of | #8821, #9827, #9828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 452);