Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator, prime radical | C23 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,8,12,9,11,13,19)(7,10,15,17,14,18,16,20);(5,7,13,15,9,14,8,16)(6,10,12,20,11,18,19,17);(1,2,3,4)Polycyclic presentation:
2^7|a^2=f,b^2=dfg,c^2=e,f^2=g,b^a=bd,c^a=ce,d^a=dg,d^b=dg,f^b=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #8470, #8473, #9251, … |
| maximal normal subgroup of | #8470, #8473, #9251, … |
| maximal quotient of | #9650, #9654, #9686, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 698);