Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = #72237528 (order 1024) · Inn(G) = C22⋊C4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22⋊C4 |
| commutator | C22 | C22×C8 |
| Frattini, cosocle | C2×C8 | C23 |
| socle | C2 | C2×C22⋊C8 |
| prime radical | C2×#41 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9,5,10,16,24,6,8,17,21,18,22,29,31)(3,11,13,23,14,25,27,32,12,7,15,19,26,20,28,30);(1,3)(2,7)(4,15)(5,14)(6,12)(8,11)(9,23)(10,20)(13,17)(16,28)(18,26)(19,21)(22,25)(24,32)(27,29)(30,31);(3,12)(7,11)(13,15)(14,26)(19,23)(20,25)(27,28)(30,32)Polycyclic presentation:
2^7|a^2=e,e^2=f,f^2=g,b^a=bd,d^a=dg,c^b=cg,e^b=eg44 conjugacy classes — view the full character table.
| maximal subgroup of | #7207, #7208, #11483, … |
| maximal normal subgroup of | #7207, #7208, #11483, … |
| maximal quotient of | #11419, #11441, #11461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 847);