Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C24 |
| commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
580 conjugacy classes of subgroups; only the 390 normal subgroups are shown.
This group has 580 conjugacy classes of subgroups — too many to draw. Only its 390 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7)(10,13)(14,16);(2,6)(5,7);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^7|e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=cf44 conjugacy classes — view the full character table.
| Fitting subgroup of | #3459⋊C3, #11714227, #11714228, … |
| maximal subgroup of | #8317, #8399, #8401, … |
| maximal normal subgroup of | #8317, #8399, #8401, … |
| maximal quotient of | #34057, #34082, #34083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2191);