Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) = #72232804 (order 1024) · Inn(G) = C2×C22⋊C4 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C24 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9,6,10,16,23)(3,8,13,21,11,22,24,32)(5,7,15,18,12,19,26,30)(14,17,28,29,25,20,27,31);(2,7)(3,11)(4,15)(8,20)(10,19)(13,27)(14,25)(16,26)(17,22)(21,32)(24,28)(29,31);(1,3,12,25)(2,8,19,20)(4,13,26,27)(5,14,6,11)(7,17,10,22)(9,21,30,31)(15,28,16,24)(18,29,23,32)Polycyclic presentation:
2^7|a^2=e,c^2=fg,e^2=g,b^a=bd,d^a=df,c^b=cg,e^b=ef32 conjugacy classes — view the full character table.
| maximal subgroup of | #8489, #8490, C2×#1473, … |
| maximal normal subgroup of | #8489, #8490, C2×#1473, … |
| maximal quotient of | #7666, #7669, #7672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 205);