Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | Aut(G) = #72213504 (order 1024) · Inn(G) = C2×#99 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#99 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C2×Q8 | C23 |
| prime radical | C2×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,21,28)(3,11,14,29,12,25,19,31)(4,15,17,27,18,8,20,26)(7,22,24,16,23,30,32,13);(1,3)(2,7)(4,16)(5,19)(6,12)(8,25)(9,24)(10,23)(11,15)(13,18)(14,21)(17,22)(20,30)(26,31)(27,29)(28,32);(3,12)(7,23)(11,25)(13,16)(14,19)(22,30)(24,32)(29,31)Polycyclic presentation:
2^7|a^2=e,d^2=g,e^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 | #12709, #12710, #12729, … |
| maximal normal subgroup of | #12709, #12710, #12729, … |
| maximal quotient of | #11432, #11448, #11480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 865);