Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) = #72232676 (order 2048) · Inn(G) = C2×C22⋊C4 · Out(G) = C2×H(5,𝔽2) |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C24 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,17,19,24)(12,16)(13,15)(14,18,22,20)(21,23);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,20)(13,19)(15,22)(16,24)(17,23)(18,21);(10,15)(14,22)(16,23)(17,24)Polycyclic presentation:
2^7|a^2=e,b^2=f,d^2=g,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 | #8488, #8517, #8519, … |
| maximal normal subgroup of | #8488, #8517, #8519, … |
| maximal quotient of | #7593, #7595, #7600, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 243);