Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) = #72233717 (order 2048) · Inn(G) = C2×C22⋊C4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C42 | C23 |
| prime radical | C23 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
99 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8,6,9,4,13)(3,10,11,15,12,7,14,16)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,32)(29,31);(1,3,6,12)(2,7,9,10)(4,11,5,14)(8,15,13,16)(17,19,21,27)(18,22,23,31)(20,29,30,26)(24,32,28,25);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(18,23)(22,31)(24,28)(25,32)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=fg,d^2=g,e^2=fg,b^a=bd,c^a=cf,d^a=df,d^b=dg,e^b=efg32 conjugacy classes — view the full character table.
| maximal subgroup of | #8490, #8501, #8503, … |
| maximal normal subgroup of | #8490, #8501, #8503, … |
| maximal quotient of | #7581, #7583, #7611, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 286);