Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| socle | C22 | C22×D4 |
| prime radical | C23 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24);(1,3,4,8)(2,5,6,7);(9,11,13,20,14,21,12,19)(10,15,16,18,17,23,22,24);(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)Polycyclic presentation:
2^7|a^2=e,b^2=e,c^2=f,d^2=g,f^2=g,b^a=be,c^a=cf,f^a=fg44 conjugacy classes — view the full character table.
| maximal subgroup of | #9068, #9749, #9752, … |
| maximal normal subgroup of | #9068, #9749, #9752, … |
| maximal quotient of | #20797, #21151, #21318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1892);