Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
830 conjugacy classes of subgroups; only the 430 normal subgroups are shown.
This group has 830 conjugacy classes of subgroups — too many to draw. Only its 430 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 12 points:
(2,4)(5,6)(7,11)(8,9)(10,12);(6,9)(10,11);(1,2,3,4);(5,6)(7,10)(8,9)(11,12);(5,7)(6,10)(8,12)(9,11)Polycyclic presentation:
2^7|c^2=g,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| maximal subgroup of | #8350, #8353, #8428, … |
| maximal normal subgroup of | #8350, #8353, #8428, … |
| maximal quotient of | #34181, #34205, #34206, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2228);