Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) = C2×#60399 (order 512) · Inn(G) = #99 · Out(G) = C24 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #99 |
| commutator | C2×C4 | C42 |
| Frattini, prime radical, cosocle | C2×#41 | C22 |
| socle | C2 | #342 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
81 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,11,13,23)(4,15,16,8)(5,18)(6,10)(7,12)(9,19)(14,28,27,24)(17,22,30,29)(20,32)(21,26)(25,31);(1,3,5,14,6,13,19,27)(2,7,9,20,10,21,18,31)(4,12,17,25,16,26,30,32)(8,11,22,24,15,23,29,28)Polycyclic presentation:
2^7|a^2=d,b^2=e,c^2=g,e^2=g,b^a=bc,c^a=cf,e^a=eg,f^a=fg,d^b=dfg,d^c=dg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1395, #12589, #12633, … |
| maximal normal subgroup of | C2×#1395, #12589, #12633, … |
| maximal quotient of | #7190, #7205, #7206, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 127);