Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) = #1202 (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 | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | #121 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,9,21)(3,12)(4,17)(8,23,24,29)(10,22,11,18)(13,28,25,20)(15,31)(16,27)(19,30)(26,32);(1,2)(3,13)(4,18)(5,10)(6,11)(7,19)(8,16)(9,14)(12,24)(15,28)(17,21)(20,31)(22,30)(23,32)(25,27)(26,29);(1,3,14,27)(2,8,9,24)(4,15,30,31)(5,16,6,12)(7,20,21,28)(10,13,11,25)(17,32,19,26)(18,23,22,29)Polycyclic presentation:
2^7|a^2=e,c^2=fg,d^2=g,b^a=bd,c^a=cg,d^a=df,c^b=cf,d^b=dg,e^b=efg32 conjugacy classes — view the full character table.
| maximal subgroup of | #8500, #8501, #8502, … |
| maximal normal subgroup of | #8500, #8501, #8502, … |
| maximal quotient of | #7580, #7581, #7584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 253);