Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C26×#120 (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 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,9,10,7)(4,12,13,8)(11,14,16,15)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,28)(31,32);(2,6)(3,10)(8,12)(11,16)(18,23)(20,24)(26,29)(28,31);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,30)(18,26,23,29)(19,27,25,32)(20,28,24,31)Polycyclic presentation:
2^7|a^2=g,c^2=f,d^2=fg,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 | #62159, #62160, #62161, … |
| maximal normal subgroup of | #62159, #62160, #62161, … |
| maximal quotient of | #34385, #34392, #34403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2297);