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) = C2×C42 |
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 | |
208 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 84 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)(12,14)(15,16)(17,18,21,20)(19,22,23,24);(3,7)(6,8)(10,11)(14,15)(18,20)(22,24);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,14,16,15)(17,19,21,23)(18,22,20,24);(1,3)(2,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,22,23,24)Polycyclic presentation:
2^7|a^2=eg,c^2=e,d^2=e,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 | #12199, #12275, C2×#2832, … |
| maximal normal subgroup of | #12199, #12275, C2×#2832, … |
| maximal quotient of | #14625, #14632, #14638, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1564);