Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) = C22×#511771 (order 2048) · Inn(G) = #120 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,4,3)(6,9)(7,11)(10,12);(1,3,4,2)(5,6)(7,12)(8,10)(9,11);(7,11)(9,12)Polycyclic presentation:
2^7|a^2=f,b^2=f,d^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dg,e^b=eg,d^c=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7375, #12958, #12959, … |
| maximal normal subgroup of | #7375, #12958, #12959, … |
| maximal quotient of | #8835, #8837, #8853, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 588);