Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
821 conjugacy classes of subgroups; only the 255 normal subgroups are shown.
This group has 821 conjugacy classes of subgroups — too many to draw. Only its 255 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(6,8)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24);(1,2,3,4)(9,11,12,18,13,17,19,23)(10,14,15,21,16,20,22,24);(5,6,7,8);(11,17)(14,20)(18,23)(21,24)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=f,g^2=h,b^a=be,c^a=cf,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #110225, #110227, #110580, … |
| maximal normal subgroup of | #110225, #110227, #110580, … |
| maximal quotient of | #204088, #204089, #204090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14177);