Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | Aut(G) = #6354079 (order 512) · Inn(G) = #476 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | #135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,12)(3,13,16,11)(4,17)(5,21,14,10)(6,8,23,25)(9,26)(15,29)(18,31)(19,30)(20,24)(22,27)(28,32);(1,3,14,23,7,16,5,6)(2,8,21,11,12,25,10,13)(4,15,19,32,20,26,18,22)(9,24,28,30,29,17,27,31);(1,4)(2,9)(3,15)(5,18)(6,22)(7,20)(8,24)(10,27)(11,30)(12,29)(13,31)(14,19)(16,26)(17,25)(21,28)(23,32)Polycyclic presentation:
2^7|a^2=df,b^2=dg,d^2=g,e^2=f,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ef,f^a=fg,e^c=efg,f^c=fg23 conjugacy classes — view the full character table.
| maximal subgroup of | #12463, #12464, #12465, … |
| maximal normal subgroup of | #12463, #12464, #12465, … |
| maximal quotient of | #12824, #12950, #13116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 963);