Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) = #72232710 (order 2048) · Inn(G) = C2×C22⋊C4 · Out(G) = C2×H(5,𝔽2) |
| 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 | C23 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
104 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)(4,16)(8,23,24,32)(10,22,11,17)(14,29)(15,28)(18,19)(20,25,31,26)(27,30);(1,2,5,10)(3,8,15,25)(4,17,16,21)(6,11,13,9)(7,19,22,18)(12,26,28,24)(14,23,30,31)(20,29,32,27);(1,3,13,28)(2,8,9,24)(4,14,19,29)(5,15,6,12)(7,20,21,31)(10,25,11,26)(16,30,18,27)(17,23,22,32)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=fg,d^2=g,b^a=bd,c^a=cg,d^a=df,d^b=dg,e^b=efg26 conjugacy classes — view the full character table.
| maximal subgroup of | #8501, #8502, #8503, … |
| maximal normal subgroup of | #8501, #8502, #8503, … |
| maximal quotient of | #7596, #7604, #7605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 247);