Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 147456 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C23 | C24 |
| commutator | C2 | C22×C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,4,3)(5,6,8,7)(10,12)(14,15);(5,6,8,7)(9,10,13,12)(11,15,16,14);(1,3,4,2)(5,7,8,6)(9,11,13,16)(10,14,12,15);(9,12,13,10)(11,15,16,14)Polycyclic presentation:
2^7|a^2=f,b^2=g,c^2=ef,d^2=e,b^a=be,d^a=de,c^b=ce68 conjugacy classes — view the full character table.
| derived subgroup of | #73286, C2×#73286, #11709755, … |
| Fitting subgroup of | #73166, #73167, #73286, … |
| maximal subgroup of | #9324, #9326, #17549, … |
| maximal normal subgroup of | #9324, #9326, #17549, … |
| maximal quotient of | #13750, #13783, #13786, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1008);