Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) = #72234537 (order 8192) · Inn(G) = C24 · Out(G) = #6342371 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
466 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 466 conjugacy classes of subgroups — too many to draw. Only its 112 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 — 4 generators on 12 points:
(1,2,3,4)(6,8);(2,4)(10,12);(5,6)(7,8)(9,10)(11,12);(1,2)(3,4)(5,6)(7,8)Polycyclic presentation:
2^7|a^2=e,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de32 conjugacy classes — view the full character table.
| maximal subgroup of | #13346, #13422, #36702, … |
| maximal normal subgroup of | #13346, #13422, #36702, … |
| maximal quotient of | #14357, #14358, #14361, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1401);