Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C24 |
| commutator | C2 | C23×C8 |
| socle | C22 | C23×C4 |
| prime radical | C22×D4 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,7)(4,6)(5,8)(9,10,11,13,12,14,15,16);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7);(2,6)(5,7)Polycyclic presentation:
2^7|a^2=f,f^2=g,b^a=be,d^a=de,c^b=ce68 conjugacy classes — view the full character table.
| maximal subgroup of | #8835, #8851, #8957, … |
| maximal normal subgroup of | #8835, #8851, #8957, … |
| maximal quotient of | #20523, #20533, #20555, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1704);