Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
108 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)(6,10)(7,11)(8,14)(12,20)(13,18)(15,19)(16,17);(5,6,11,17,8,12,15,18,9,13,19,20,14,16,7,10);(1,2,3,4)Polycyclic presentation:
2^7|b^2=df,c^2=e,d^2=fg,f^2=g,b^a=bd,c^a=ce,d^a=df,f^a=fg38 conjugacy classes — view the full character table.
| maximal subgroup of | #7444, #7448, #12316, … |
| maximal normal subgroup of | #7444, #7448, #12316, … |
| maximal quotient of | #12503, #12509, #12515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 978);