Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,15)(14,16);(1,3)(2,6)(4,7)(5,8);(1,4,5,2)(3,7,8,6)(10,13)(11,16)(14,15);(3,8)(6,7)Polycyclic presentation:
2^7|a^2=e,c^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=de,f^c=fg35 conjugacy classes — view the full character table.
| maximal subgroup of | #12340, #12344, #12392, … |
| maximal normal subgroup of | #12340, #12344, #12392, … |
| maximal quotient of | #22485, #22611, D4×#107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2066);