Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = #72268601 (order 4096) · Inn(G) = C25 · Out(G) = C22×H(5,𝔽2) |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23);(2,5)(6,8)(10,14)(11,18)(16,20)(19,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24);(1,3,4,7)(2,6,5,8)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23);(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)Polycyclic presentation:
2^7|a^2=f,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df38 conjugacy classes — view the full character table.
| maximal subgroup of | #8484, #8511, #18242, … |
| maximal normal subgroup of | #8484, #8511, #18242, … |
| maximal quotient of | #34377, #34382, #34399, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2293);