Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | Aut(G) = #72215858 (order 1024) · Inn(G) = #120 · Out(G) = #120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator, Frattini, cosocle | C42 | C23 |
| prime radical | C2×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,15,19)(11,20,22,23)(12,21,16,24)(13,18,14,17);(3,7)(5,8)(9,11)(10,16)(12,19)(13,24)(14,21)(15,22)(17,23)(18,20);(1,2)(3,8)(4,6)(5,7)(9,12,15,16)(10,11,19,22)(13,23,14,20)(17,21,18,24)Polycyclic presentation:
2^7|a^2=f,c^2=f,d^2=f,e^2=fg,b^a=bd,c^a=ce,d^a=df,e^a=efg,d^b=df,e^b=eg,d^c=dg,e^c=efg20 conjugacy classes — view the full character table.
| maximal subgroup of | #13680, C2×#1660, #18984, … |
| maximal normal subgroup of | #13680, C2×#1660, #18984, … |
| maximal quotient of | #7751, #7754, #7757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 392);