Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | Aut(G) = #499787 (order 512) · Inn(G) = B(4,𝔽2) · Out(G) = D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C23 |
| prime radical | #454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ 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)(3,12)(4,16)(5,8)(6,11)(7,13)(9,19)(10,21)(14,29)(15,27)(17,25)(18,26)(20,30)(22,28)(23,31)(24,32);(1,3,6,15)(2,7,11,25)(4,17,19,13)(5,20,21,14)(8,23,10,24)(9,27,16,12)(18,31,22,32)(26,29,28,30);(2,8)(5,21)(7,23)(9,28)(10,11)(12,29)(14,20)(16,26)(18,22)(24,25)(27,30)(31,32)Polycyclic presentation:
2^7|b^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,d^b=dg,e^b=efg,d^c=df,e^c=eg,e^d=eg17 conjugacy classes — view the full character table.
| maximal subgroup of | #13679, #13680, C2×#2203, … |
| maximal normal subgroup of | #13679, #13680, C2×#2203, … |
| maximal quotient of | #12097, #12098, #12126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 935);