Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D8 |
| commutator | C8 | C22×C4 |
| Frattini, cosocle | C2×C8 | C23 |
| prime radical | C23 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
110 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)(7,16)(8,13)(9,22)(10,26)(11,15)(12,18)(14,24)(17,30)(19,36)(20,33)(21,31)(23,29)(25,32)(27,34)(28,35);(5,7,11,20)(6,12,15,28)(8,19,10,17)(9,23,25,34)(13,29,26,27)(14,30,31,36)(16,32,33,22)(18,21,35,24);(5,8,11,10)(6,13,15,26)(7,17,20,19)(9,21,25,24)(12,27,28,29)(14,22,31,32)(16,30,33,36)(18,34,35,23)Polycyclic presentation:
2^7|a^2=e,b^2=g,c^2=g,d^2=fg,f^2=g,b^a=bd,d^a=df,f^a=fg,c^b=cg,d^b=df,f^b=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #12648, #12742, #12751, … |
| maximal normal subgroup of | #12648, #12742, #12751, … |
| maximal quotient of | #11594, #11650, #11662, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 873);