Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C24 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
215 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,10)(7,9,11,12)(14,18)(19,21)(20,24)(27,28);(2,4)(7,11)(9,12)(13,14)(15,21)(16,24)(17,18)(19,23)(20,26)(22,27)(25,28);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,23)(14,19,18,21)(16,25,26,22)(20,28,24,27);(5,7)(6,9)(8,11)(10,12)(13,16,17,26)(14,20,18,24)(15,22,23,25)(19,27,21,28)Polycyclic presentation:
2^7|a^2=f,c^2=e,d^2=e,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de26 conjugacy classes — view the full character table.
| maximal subgroup of | #18463, #18465, #18470, … |
| maximal normal subgroup of | #18463, #18465, #18470, … |
| maximal quotient of | #14625, #14626, #14628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1553);