Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) = #72234537 (order 8192) · Inn(G) = C24 · Out(G) = #6342371 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(17,18,21,24)(19,23,20,22);(2,5)(6,7)(9,10,12,14)(11,13,15,16)(17,19,21,20)(18,22,24,23);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,19)(18,23,24,22);(1,3,4,8)(2,6,5,7)(9,11,12,15)(10,13,14,16)(17,18)(19,22)(20,23)(21,24)Polycyclic presentation:
2^7|a^2=f,b^2=eg,c^2=f,d^2=g,b^a=be,c^a=cf,d^a=de,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | #12182, #12258, C2×#2612, … |
| maximal normal subgroup of | #12182, #12258, C2×#2612, … |
| maximal quotient of | #13963, #14087, #14088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1344);