Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 4096 |
| 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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
193 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(11,17)(12,18)(14,20)(16,21)(19,23)(22,24);(2,5)(6,8)(9,10,13,15)(11,14,19,22)(12,16,18,21)(17,20,23,24);(9,11,18,23,13,19,12,17)(10,14,21,24,15,22,16,20);(1,3,4,7)(2,6,5,8)(10,15)(14,22)(16,21)(20,24)Polycyclic presentation:
2^7|b^2=g,c^2=fg,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | #11770, #11913, #11989, … |
| maximal normal subgroup of | #11770, #11913, #11989, … |
| maximal quotient of | #20821, #21282, #21290, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1908);