Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | Aut(G) = #6346637 (order 512) · Inn(G) = #447 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | #575 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10)(3,11,13,29)(4,15,17,31)(5,20,18,9)(7,22,23,12)(8,25,27,19)(14,30,21,26)(16,28,32,24);(1,3)(2,7)(4,16)(5,21)(6,13)(8,26)(9,28)(10,23)(11,15)(12,19)(14,18)(17,32)(20,24)(22,25)(27,30)(29,31);(4,17)(8,27)(11,29)(12,22)(15,31)(16,32)(19,25)(26,30)Polycyclic presentation:
2^7|a^2=g,d^2=fg,f^2=g,b^a=bd,c^a=ce,d^a=df,f^a=fg,d^b=df,e^b=eg,f^b=fg,d^c=dg20 conjugacy classes — view the full character table.
| maximal subgroup of | #12115, #12116, #12118, … |
| maximal normal subgroup of | #12115, #12116, #12118, … |
| maximal quotient of | #12105, #12106, #12107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 923);