Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) = #72245616 (order 6144) · Inn(G) = C22×D4 · Out(G) = C23×S4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D4 |
| commutator | C4 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| socle | C2 | C22×C4⋊C4 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,20)(3,12,15,22,16,26,14,8)(4,9,19,24,17,23,18,30)(13,25,29,31,27,32,28,21);(1,3,5,14,7,16,6,15)(2,8,10,12,11,22,20,26)(4,13,18,28,17,27,19,29)(9,21,24,25,23,31,30,32);(1,4)(2,9)(3,13)(5,18)(6,19)(7,17)(8,21)(10,24)(11,23)(12,25)(14,28)(15,29)(16,27)(20,30)(22,31)(26,32);(4,17)(9,23)(13,27)(18,19)(21,31)(24,30)(25,32)(28,29)Polycyclic presentation:
2^7|a^2=f,b^2=e,e^2=g,f^2=g,b^a=be,e^a=eg,d^c=dg44 conjugacy classes — view the full character table.
| Fitting subgroup of | #2916⋊C3, #11711551, #11711554, … |
| maximal subgroup of | #12877, #12878, #12927, … |
| maximal normal subgroup of | #12877, #12878, #12927, … |
| maximal quotient of | #17788, #17789, #17798, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1648);