Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C210 (order 1024) · Inn(G) = C25 · Out(G) = C25 |
| 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 | |
477 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,8)(3,13)(4,18)(5,21)(9,26)(10,16)(12,25)(14,24)(17,30)(19,27)(20,31)(22,32);(1,2)(3,14)(4,19)(5,11)(6,8)(7,12)(9,13)(10,20)(15,21)(16,18)(17,26)(22,28)(23,25)(24,30)(27,31)(29,32);(1,3,6,17)(2,9,8,24)(4,15,20,11)(5,16,21,27)(7,13,23,30)(10,22,19,32)(12,14,25,26)(18,29,31,28);(1,4)(2,10)(3,15)(5,14)(6,20)(7,18)(8,19)(9,22)(11,17)(12,27)(13,29)(16,25)(21,26)(23,31)(24,32)(28,30);(1,5,7,22)(2,11,12,28)(3,16,13,19)(4,14,18,9)(6,21,23,32)(8,15,25,29)(10,17,27,30)(20,26,31,24)Polycyclic presentation:
2^7|c^2=f,e^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 | C2×#3566, #62743, #62755, … |
| maximal normal subgroup of | C2×#3566, #62743, #62755, … |
| maximal quotient of | #34387, #34395, #34400, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2298);