Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) = #72234537 (order 8192) · Inn(G) = C24 · Out(G) = #6342371 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C24 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
198 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 198 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)(9,10,13,14)(11,12,15,16)(19,23)(21,24);(3,7)(6,8)(10,14)(12,16)(17,18,20,22)(19,24,23,21);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,19,20,23)(18,21,22,24);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,14)(13,16)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^7|a^2=e,b^2=g,c^2=g,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de26 conjugacy classes — view the full character table.
| maximal subgroup of | #12125, #12219, C2×#2841, … |
| maximal normal subgroup of | #12125, #12219, C2×#2841, … |
| maximal quotient of | #14635, #14636, #14637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1573);