Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) = #72233546 (order 4096) · Inn(G) = C2×C22⋊C4 · Out(G) = C2×B(4,𝔽2) |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C42 | C23 |
| prime radical | #532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10)(11,18,20,23)(12,21)(13,16)(14,17)(15,19,24,22);(2,5)(3,7)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)Polycyclic presentation:
2^7|a^2=e,c^2=g,d^2=g,e^2=f,b^a=bd,c^a=cg,d^a=df,d^b=dg,e^b=efg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7484, #7494, #8475, … |
| maximal normal subgroup of | #7484, #7494, #8475, … |
| maximal quotient of | #7597, #7603, #7605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 268);