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, cosocle | C23 | C24 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
220 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,8)(9,10)(11,13)(12,15)(14,16)(17,18,21,24)(19,23,20,22);(10,11)(14,15)(17,19,21,20)(18,22,24,23);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,22)(20,23)(21,24);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^7|a^2=eg,b^2=eg,c^2=g,d^2=f,b^a=be,c^a=cf,d^a=de,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | #12196, #12272, C2×#2593, … |
| maximal normal subgroup of | #12196, #12272, C2×#2593, … |
| maximal quotient of | #13928, #14031, #14037, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1325);