Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | Aut(G) = #6346637 (order 512) · Inn(G) = #481 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | C23 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
62 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)(3,11,12,29)(4,15,16,32)(5,20,18,9)(7,22,23,13)(8,25,26,19)(14,27,21,30)(17,24,31,28);(4,16)(8,26)(11,29)(13,22)(15,32)(17,31)(19,25)(27,30);(1,3,5,21,6,12,18,14)(2,7,9,28,10,23,20,24)(4,17,19,22,16,31,25,13)(8,27,15,29,26,30,32,11)Polycyclic presentation:
2^7|a^2=g,c^2=df,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,e^b=eg,d^c=dg,e^c=efg,f^c=fg20 conjugacy classes — view the full character table.
| maximal subgroup of | #12299, #12300, #12302, … |
| maximal normal subgroup of | #12299, #12300, #12302, … |
| maximal quotient of | #12875, #12931, #13009, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 971);