Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,23)(22,24);(9,11,12,18,13,19,21,24)(10,14,16,17,15,22,20,23);(1,2,3,4)(10,15)(14,22)(16,20)(17,23)Polycyclic presentation:
2^7|a^2=f,b^2=d,c^2=eg,d^2=g,b^a=bd,c^a=ce,d^a=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #7271, #7275, #7279, … |
| maximal normal subgroup of | #7271, #7275, #7279, … |
| maximal quotient of | #8807, #8811, #8814, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 578);