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 | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(9,10)(11,18)(12,15)(13,21)(14,17)(16,22)(19,23)(20,24);(2,5)(6,8)(9,11,14,12)(10,15,17,18)(13,19,22,20)(16,23,21,24);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,15,17,18)(13,20,22,19)(16,23,21,24);(1,3,4,7)(2,6,5,8)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)Polycyclic presentation:
2^7|b^2=f,c^2=f,d^2=g,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 | #8486, #8513, #18244, … |
| maximal normal subgroup of | #8486, #8513, #18244, … |
| maximal quotient of | #34378, #34382, #34398, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2291);