Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C22 |
| commutator | C2 | C43 |
| Frattini, cosocle | C22×C4 | C23 |
| socle, prime radical | C23 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(1,2,3,4)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(5,6,7,8)(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=f,d^2=g,c^b=cg80 conjugacy classes — view the full character table.
| derived subgroup of | #72283798, #72283823, #72283827, … |
| Frattini subgroup of | H(3,ℤ/8), #93253, #93254, … |
| Fitting subgroup of | #1726⋊C3, #11710953, #11711007, … |
| maximal subgroup of | #7150, #8137, #8138, … |
| maximal normal subgroup of | #7150, #8137, #8138, … |
| maximal quotient of | C8×#97, #7841, #7842, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 458);