Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | Aut(G) = C2×#72218857 (order 3072) · Inn(G) = #120 · Out(G) = C22×S4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #120 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #379 |
| prime radical | #585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,9,18,23)(3,11,14,24,13,7,17,20)(4,8,16,21,15,22,28,32)(12,19,27,30,26,25,29,31);(1,3)(2,7)(4,12)(5,17)(6,13)(8,19)(9,11)(10,24)(14,18)(15,26)(16,29)(20,23)(21,31)(22,25)(27,28)(30,32);(1,4)(2,8)(3,12)(5,16)(6,15)(7,19)(9,22)(10,21)(11,25)(13,26)(14,27)(17,29)(18,28)(20,30)(23,32)(24,31)Polycyclic presentation:
2^7|a^2=f,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^b=eg,f^b=fg,d^c=dg32 conjugacy classes — view the full character table.
| Fitting subgroup of | #1791⋊C3, #11711587, #11711589, … |
| maximal subgroup of | #13045, #13046, #13051, … |
| maximal normal subgroup of | #13045, #13046, #13051, … |
| maximal quotient of | #8316, #8317, #8348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 523);