Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = C2×#1202 (order 2048) · Inn(G) = C24 · Out(G) = C2×D42 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C24 |
| commutator, socle | C22 | C23×C4 |
| prime radical | C23 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ 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)(3,8)(4,6)(5,7)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(1,3,5,4)(2,6,7,8);(1,4,5,3)(2,6,7,8)(9,11,12,18,13,17,19,23)(10,14,15,21,16,20,22,24);(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^7|a^2=f,b^2=e,c^2=ef,f^2=g,b^a=be,d^c=dg50 conjugacy classes — view the full character table.
| maximal subgroup of | #8931, #8933, #9054, … |
| maximal normal subgroup of | #8931, #8933, #9054, … |
| maximal quotient of | #20439, #20441, #20519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1694);