Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C22×#6342371 (order 2048) · Inn(G) = C25 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| prime radical | #538 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,10)(3,11,12,8)(4,15,18,9)(5,7,17,21)(13,23,25,24)(14,22,26,19)(16,29,30,20)(27,32,28,31);(2,7)(3,12)(4,16)(8,22)(10,21)(11,19)(13,27)(14,26)(18,30)(23,24)(25,28)(31,32);(1,3,5,14)(2,8,7,19)(4,13,16,28)(6,12,17,26)(9,23,20,31)(10,11,21,22)(15,24,29,32)(18,25,30,27);(1,4,17,30)(2,9,21,29)(3,13,26,27)(5,16,6,18)(7,20,10,15)(8,23,22,32)(11,24,19,31)(12,25,14,28);(2,7)(8,19)(9,20)(10,21)(11,22)(15,29)(23,31)(24,32)Polycyclic presentation:
2^7|a^2=g,c^2=f,d^2=fg,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 | #8490, #18250, #18251, … |
| maximal normal subgroup of | #8490, #18250, #18251, … |
| maximal quotient of | #34381, #34389, #34401, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2296);