Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
565 conjugacy classes of subgroups; only the 299 normal subgroups are shown.
This group has 565 conjugacy classes of subgroups — too many to draw. Only its 299 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:
(1,2,4,3)(9,10)(11,18)(12,14)(13,16)(15,19)(17,21)(20,23)(22,24);(5,6,8,7)(10,14)(12,21)(15,22)(16,17)(18,23)(20,24);(5,7,8,6)(9,11)(10,15)(12,20)(13,19)(14,22)(16,18)(17,23)(21,24);(1,3,4,2)(10,16)(11,19)(14,17)(20,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=f,e^2=h,b^a=be,c^a=ch,d^a=dh,e^a=eh,e^b=eh,d^c=dh88 conjugacy classes — view the full character table.
| Fitting subgroup of | #20506⋊C3 |
| maximal subgroup of | #221350, #221352, #221355, … |
| maximal normal subgroup of | #221350, #221352, #221355, … |
| maximal quotient of | #183880, #183904, #183916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13494);