Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) = #72252166 (order 4096) · Inn(G) = #121 · Out(G) = #2403 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
171 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,16)(7,17,21,28)(8,22,26,31)(9,15,10,14)(12,20,29,19)(13,25,32,24)(18,33,35,27)(23,30,34,36);(2,4)(5,7)(6,12)(8,23)(9,20)(10,19)(11,21)(13,30)(14,28)(15,17)(16,29)(18,25)(22,33)(24,35)(26,34)(27,31)(32,36);(5,8,11,26)(6,13,16,32)(7,18,21,35)(9,25,10,24)(12,27,29,33)(14,31,15,22)(17,30,28,36)(19,23,20,34)Polycyclic presentation:
2^7|a^2=g,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,d^b=dg,f^b=fg,d^c=dg,e^c=eg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7399, #7401, #12111, … |
| maximal normal subgroup of | #7399, #7401, #12111, … |
| maximal quotient of | #9976, #9979, #9989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 752);