Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) = #72231721 (order 2048) · Inn(G) = C22⋊C4 · Out(G) = #2846 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22⋊C4 |
| commutator | C2×C4 | C42 |
| Frattini, cosocle | C2×C42 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4)(6,10)(7,8)(11,12)(13,14)(15,17)(16,19)(18,20);(5,6,8,11,9,12,7,10)(14,17)(15,20)(18,19)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=f,e^2=fg,b^a=bc,c^a=cf,e^a=efg,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | #7417, #7443, C2×#1296, … |
| maximal normal subgroup of | #7417, #7443, C2×#1296, … |
| maximal quotient of | #7025, #7031, #7039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 28);