Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) = #72234537 (order 8192) · Inn(G) = C24 · Out(G) = #6342371 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C24 |
| commutator | C22 | C23×C4 |
| prime radical | C24 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
264 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(13,14,17,18)(15,20,19,16);(1,3,4,2)(6,9)(10,11)(14,18)(15,19);(5,6)(7,10)(8,9)(11,12)(13,15,17,19)(14,16,18,20);(5,7)(6,10)(8,12)(9,11)(13,16)(14,19)(15,18)(17,20)Polycyclic presentation:
2^7|a^2=g,b^2=fg,c^2=f,b^a=be,c^a=cf,d^a=de,c^b=ce44 conjugacy classes — view the full character table.
| maximal subgroup of | #8867, #8868, #8871, … |
| maximal normal subgroup of | #8867, #8868, #8871, … |
| maximal quotient of | #13778, #13782, #13792, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1061);