Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | Aut(G) = #72238017 (order 2048) · Inn(G) = C2×#99 · Out(G) = #120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#99 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C2×D4 | C23 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
119 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)(3,11,14,26)(4,8)(5,18,20,9)(6,10)(7,13)(12,27,15,29)(16,31,32,24)(17,25)(19,22)(21,28)(23,30);(1,3,6,14)(2,7,10,22)(4,15,17,12)(5,19,20,13)(8,23,25,21)(9,26,18,11)(16,28,32,30)(24,27,31,29);(1,4,6,17)(2,8,10,25)(3,12,14,15)(5,16,20,32)(7,21,22,23)(9,24,18,31)(11,27,26,29)(13,28,19,30)Polycyclic presentation:
2^7|a^2=e,b^2=g,c^2=g,d^2=g,b^a=bd,d^a=df,f^a=fg,c^b=cg,d^b=dg,e^b=efg,e^d=eg23 conjugacy classes — view the full character table.
| maximal subgroup of | #12712, #12713, #12721, … |
| maximal normal subgroup of | #12712, #12713, #12721, … |
| maximal quotient of | #11434, #11449, #11482, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 861);