Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) = #6442738 (order 512) · Inn(G) = #117 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #117 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C2×C8 | C23 |
| socle | C2 | #432 |
| prime radical | C23 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #97 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
61 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,10,5,11,18,27,6,9,17,25,19,8,22,24)(3,12,14,26,15,23,29,32,13,7,16,21,28,20,30,31);(1,3)(2,7)(4,16)(5,15)(6,13)(8,23)(9,12)(10,26)(11,20)(14,17)(18,30)(19,28)(21,25)(22,29)(24,31)(27,32);(2,8,9,11)(3,13)(4,17)(7,20,12,23)(10,27,25,24)(15,28)(18,22)(21,32,26,31)Polycyclic presentation:
2^7|a^2=e,c^2=d,e^2=f,f^2=g,b^a=bd,c^a=cf,d^a=dg,c^b=cg,e^b=eg,e^c=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7239, #7240, #11537, … |
| maximal normal subgroup of | #7239, #7240, #11537, … |
| maximal quotient of | #12596, #12608, #12660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 898);