Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) = #72234840 (order 1024) · Inn(G) = #99 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #99 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C2×D4 | C23 |
| socle | C2 | C2×#99 |
| prime radical | #532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,7)(4,8,12,6)(5,9)(10,14,16,13)(11,15);(2,6)(4,12)(7,13)(8,9)(10,16)(14,15);(1,3,5,11)(2,7,9,15)(4,10,12,16)(6,13,8,14)Polycyclic presentation:
2^7|a^2=e,c^2=g,d^2=g,b^a=bd,d^a=df,f^a=fg,d^b=dg,e^b=efg,e^d=eg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7482, #7486, #7490, … |
| maximal normal subgroup of | #7482, #7486, #7490, … |
| maximal quotient of | #11431, #11447, #11476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 858);