Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | Aut(G) = #72231230 (order 2048) · Inn(G) = #95 · Out(G) = #392 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #95 |
| commutator | C2×C4 | C42 |
| Frattini, cosocle | C2×C42 | C22 |
| prime radical | C23 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,12,17,14,18,13,16)(11,19,20,15,22,23,21,24);(2,5)(3,7)(9,11,13,21,14,22,12,20)(10,15,17,23,18,24,16,19)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=fg,d^2=f,e^2=fg,b^a=bc,c^a=cf,e^a=efg,d^b=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7080, #7081, #7082, … |
| maximal normal subgroup of | #7080, #7081, #7082, … |
| maximal quotient of | #7020, #7023, #7038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 22);