Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
127 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,8,10)(7,11,12,9)(13,14,18,20,17,15,16,19);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,17,14)(16,19,18,20);(2,4)(13,14,17,15)(16,20,18,19)Polycyclic presentation:
2^7|a^2=dg,b^2=de,c^2=g,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 | #7437, #7801, #7802, … |
| maximal normal subgroup of | #7437, #7801, #7802, … |
| maximal quotient of | #10663, #10670, #10675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 828);