Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | Aut(G) = D4×#442 (order 512) · Inn(G) = #447 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | #578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,8,14)(3,15,19,43)(4,20,23,40)(5,24,28,38)(6,13,7,12)(9,31,34,17)(10,29,36,22)(11,26,39,27)(16,44,48,61)(18,49,30,37)(21,52,55,35)(25,41,56,33)(32,53,60,64)(42,47,63,46)(45,50,58,59)(51,57,62,54);(1,3)(2,9)(4,16)(5,25)(6,18)(7,30)(8,19)(10,32)(11,37)(12,33)(13,41)(14,34)(15,24)(17,27)(20,50)(21,53)(22,46)(23,48)(26,31)(28,56)(29,47)(35,61)(36,60)(38,43)(39,49)(40,59)(42,51)(44,52)(45,54)(55,64)(57,58)(62,63);(1,4,8,23)(2,10,14,36)(3,16,19,48)(5,21,28,55)(6,29,7,22)(9,32,34,60)(11,35,39,52)(12,40,13,20)(15,42,43,63)(17,45,31,58)(18,47,30,46)(24,51,38,62)(25,53,56,64)(26,57,27,54)(33,59,41,50)(37,61,49,44)Polycyclic presentation:
2^7|a^2=g,c^2=g,d^2=fg,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fg,d^b=df,f^b=fg,e^c=eg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2195, #33538, #33540, … |
| maximal normal subgroup of | C2×#2195, #33538, #33540, … |
| maximal quotient of | #12112, #12113, #12168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 927);