Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 32768 |
| 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 | |
256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(6,8)(7,11)(13,14,17,19)(15,16,18,20);(5,6,9,8)(7,10,11,12)(13,15)(14,16)(17,18)(19,20);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,16)(14,18)(15,19)(17,20);(5,8,9,6)(7,12,11,10)Polycyclic presentation:
2^7|a^2=g,b^2=e,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | #9460, #9953, #9961, … |
| maximal normal subgroup of | #9460, #9953, #9961, … |
| maximal quotient of | #13881, #13904, #13972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1235);