Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
721 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 721 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,21)(15,19,17,25)(16,23,24,27)(20,26,28,22);(1,2)(3,4)(6,9)(10,12)(13,15,18,17)(14,19,21,25)(16,22,24,26)(20,27,28,23);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16,18,24)(14,20,21,28)(15,22,17,26)(19,27,25,23);(7,11)(10,12)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^c=de,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #246166, #246168, #246206, … |
| maximal normal subgroup of | #246166, #246168, #246206, … |
| maximal quotient of | #326966, #326975, #326983, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22587);