Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
761 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 761 conjugacy classes of subgroups — too many to draw. Only its 179 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)(3,4)(6,7)(9,10)(11,19)(12,16)(13,17)(14,18)(15,22)(20,23)(21,24);(5,6)(7,8)(9,10)(11,20)(12,16)(13,17)(14,18)(15,21)(19,23)(22,24);(2,4)(5,7,8,6)(9,11)(10,15)(12,21)(13,22)(14,23)(16,19)(17,20)(18,24);(9,12,14,13)(10,16,18,17)(11,21,23,22)(15,19,24,20)Polycyclic presentation:
2^8|c^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh,f^c=fh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #99712, #99713, #165697, … |
| maximal normal subgroup of | #99712, #99713, #165697, … |
| maximal quotient of | #238435, #238436, #238437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16492);