Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#548 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C24 |
| commutator, Frattini, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
810 conjugacy classes of subgroups; only the 444 normal subgroups are shown.
This group has 810 conjugacy classes of subgroups — too many to draw. Only its 444 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 14 points:
(1,2);(4,6)(8,11)(12,13);(7,8,10,11)(9,13,14,12);(3,4)(5,6);(7,9,10,14)(8,12,11,13)50 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#548⋊C3, #11711028, #11711029, … |
| maximal subgroup of | #8366, #8369, #8377, … |
| maximal normal subgroup of | #8366, #8369, #8377, … |
| maximal quotient of | C2×Q8×C22⋊C4, C2×#2341, C2×#2351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2197);