Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
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 | |
222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,9)(10,12)(13,14,18,16)(15,21,23,19)(17,20,26,24)(22,28,25,27);(2,4)(6,9)(7,11)(13,15,18,23)(14,19,16,21)(17,25,26,22)(20,28,24,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,19,23,21)(17,20,26,24)(22,28,25,27);(5,7)(6,10)(8,11)(9,12)(13,17,18,26)(14,20,16,24)(15,22,23,25)(19,27,21,28)Polycyclic presentation:
2^7|a^2=e,b^2=e,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 | #9982, #9985, #12095, … |
| maximal normal subgroup of | #9982, #9985, #12095, … |
| maximal quotient of | #14626, #14627, #14629, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1549);