Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
137 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 137 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 3 generators on 20 points:
(5,6,8,10)(7,11,12,9)(13,14,15,19,17,20,16,18);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,18)(15,16)(19,20);(2,4)(14,19)(15,16)(18,20)Polycyclic presentation:
2^7|a^2=d,b^2=de,d^2=g,e^2=g,b^a=bd,c^a=ce,c^b=cf,d^b=dg,e^b=eg,d^c=dg,e^c=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2092, #23730, #23731, … |
| maximal normal subgroup of | C2×#2092, #23730, #23731, … |
| maximal quotient of | #10659, #10665, #10671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 824);