Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) = #72240983 (order 3072) · Inn(G) = C24 · Out(G) = C23×S4 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C8 | C24 |
| commutator, socle | C2 | C23×C8 |
| prime radical | C2×#41 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,9,5,10,14,22,6,8,15,20,16,21,26,30)(3,7,12,18,13,19,25,29,11,17,23,27,24,28,31,32);(1,3)(2,7)(4,12)(5,13)(6,11)(8,17)(9,18)(10,19)(14,25)(15,23)(16,24)(20,27)(21,28)(22,29)(26,31)(30,32);(3,11)(7,17)(12,23)(13,24)(18,27)(19,28)(25,31)(29,32);(2,8)(7,17)(9,20)(10,21)(18,27)(19,28)(22,30)(29,32)Polycyclic presentation:
2^7|a^2=e,e^2=f,f^2=g,d^a=dg,c^b=cg68 conjugacy classes — view the full character table.
| Fitting subgroup of | #3407⋊C3, #11711491, #11711535, … |
| maximal subgroup of | #12618, #12833, #12915, … |
| maximal normal subgroup of | #12618, #12833, #12915, … |
| maximal quotient of | #33324, #33326, #33333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2139);