Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C24 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,16)(11,17,18,24)(12,20,22,15)(14,21,23,19);(1,3)(2,5)(4,8)(6,7)(11,18)(14,23)(17,24)(19,21);(3,8)(5,7)(9,11,13,18)(10,14,16,23)(12,21,22,19)(15,24,20,17);(2,6)(5,7)Polycyclic presentation:
2^7|a^2=g,c^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ceg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | #9342, #9935, #9937, … |
| maximal normal subgroup of | #9342, #9935, #9937, … |
| maximal quotient of | #21765, #21942, #22050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1927);