Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C23 |
| commutator, socle | C23 | C22×C4 |
| prime radical | C22×#41 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(2,4)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(6,8)(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^7|a^2=f,f^2=g,b^a=bd,c^a=ce,c^b=cg44 conjugacy classes — view the full character table.
| Frattini subgroup of | #93202, #93204, #93205, … |
| Fitting subgroup of | #1780⋊C3, #11710945, #11711138, … |
| maximal subgroup of | #7058, #7305, #7306, … |
| maximal normal subgroup of | #7058, #7305, #7306, … |
| maximal quotient of | #8304, #8305, #8306, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 512);